diff options
Diffstat (limited to '16449-h')
| -rw-r--r-- | 16449-h/16449-h.htm | 10656 |
1 files changed, 10656 insertions, 0 deletions
diff --git a/16449-h/16449-h.htm b/16449-h/16449-h.htm new file mode 100644 index 0000000..dde900b --- /dev/null +++ b/16449-h/16449-h.htm @@ -0,0 +1,10656 @@ +<!DOCTYPE html + PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" + "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> + +<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> +<head> +<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" /> +<title>The Number Concept - Its Origin And Development</title> +<style type="text/css"> +/* leave room for page numbers; */ +body { + margin-left: 9%; + margin-right: 9%; +} + + +hr { + width: 25%; + height: 0px; + margin: 1.5em auto; + clear: both +} +hr.front { + width: 100%; + margin: 3em 0; +} +hr.deco { + width: 10%; + margin: -2em auto; +} + + +p { + text-align: justify; + text-indent: 1em; + margin: 0; +} +p.front { + text-align: center; + text-indent:0; + margin: 4em 0; +} +p.sign { + font-size: 90%; + margin: 1em 0; +} + +.sc, .initial { + font-variant: small-caps; +} +.noindent { + text-indent: 0; +} +.ctr { + text-align: center; + text-indent: 0; +} +blockquote p { text-indent: -2em; padding-left: 2em; } + +ul.toc { + list-style: none; +} +ul.toc li { + font-variant: small-caps; + position: relative; +} +ul.toc li i { + font-variant: normal; +} +ul.toc li.chapter { + text-transform: uppercase; + font-size: 120%; + text-align: center; + margin-top: 1em; +} + +ul.index { + list-style: none; +} +ul.index li { + text-indent: -1em; + padding-left: 1em; +} +ul.index li span { + padding-left: 1ex; +} +ul.index li.newletter { + margin-top: 1em; +} + +h1, h2, h3 { + text-align: center; + text-transform: uppercase; +} +h1 { + margin: 6em 0 2em 0; +} +h2 { + margin: 4em 0 2em 0; +} +h3 { + font-size: 90%; + font-weight: normal; + margin: 3em 0 3em 0; +} +h2 + h3 { + margin: -2em 0 2em 0; +} +h3 i { + text-transform: none; +} + +/* Page numbers, placed into the left margin; this style hidden from IE6.0! */ +html > * .pgnum { + position: absolute; + right: 1%; + font-size: 8pt; + text-indent: 0; + font-style: normal; + font-weight: normal; + text-align: right; + text-transform: none; + font-variant: normal; + letter-spacing:0; + visibility: visible; +} +.pgnum:before { + content: attr(title); +} + +.toc a { + position: absolute; + right: 0; +} + +div.poem { + margin: 1em 4em; + font-size: 90%; +} +.poem p { + text-align: left; + text-indent: -2em; + padding-left: 2em; + margin: 0; +} + +sup { + font-weight: normal; + font-size: 90%; + line-height: 1%; +} + +div.footnotes p { + font-size: 90%; + margin: 0.3em 3em; +} +div.footnotes p sup { + margin-right: 1ex; +} +a { + text-decoration: none; +} + +.twoline { + font-size: 200%; + padding:0; +} + + +table { + margin: 2em auto; + border-collapse: collapse; + font-size: 90%; +} +table * { + margin: 0; +} +th, td { + padding: 0 0.5ex; + vertical-align: top; +} +td.alt { + padding-left: 1em; +} + +table.border { + border: medium double black; +} +table.border td { + border: none; + border-right: thin solid black; + border-left: thin solid black; +} +table.border thead th { + border: none; + border-right: thin solid black; + border-left: thin solid black; + border-bottom: thin solid black; +} + +table#saltees td:first-child + td { + text-align: right; +} + +table#nine-new td:first-child + td { + padding-right: 1em; +} + + +table.grammar td { + width: 10em; +} +table.grammar td:first-child { + width: 4em; +} +table.grammar td:first-child + td { + width: 8em; +} +table.grammar td + td + td { + width: 14em; +} + +table.layout { + margin: 0; + border: none; + font-size: 100%; +} +table.layout td { + border: none; + margin:0; padding:0; + vertical-align: middle; +} + +th { + font-weight: normal; + text-align: left; +} +thead th { + font-weight: bold; + text-align: center; +} + + +td.indent { + text-indent: 1ex; +} + +table.scale td:first-child, +td.number +{ + text-align: right; +} + +td + td.number { + padding-left: 1em; +} + + +td { + line-height: 100%; + padding-top: 0.16em; + padding-bottom: 0.16em; +} + +caption, th.caption { + caption-side: top; + font-weight: bold; + text-align: center; + line-height: 200%; + margin: 2em auto -1em auto; +} +caption { + white-space: pre; +} +caption * { + font-weight: normal; +} + + +.lefthalf { + width: 49%; + margin-right: 1%; + float:left; +} +.righthalf { + width: 49%; + margin-left: 1%; + float:right; +} +.righthalf + * { clear: both } + +</style> +</head> +<body> + + +<pre> + +The Project Gutenberg EBook of The Number Concept, by Levi Leonard Conant + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: The Number Concept + Its Origin and Development + +Author: Levi Leonard Conant + +Release Date: August 5, 2005 [EBook #16449] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE NUMBER CONCEPT *** + + + + +Produced by Jonathan Ingram, Hagen von Eitzen and the +Online Distributed Proofreading Team at https://www.pgdp.net + + + + + + +</pre> + + +<p class="front"><a class="pgnum" id="page-ii" title="- ii -"></a>THE MACMILLAN COMPANY<br/> +<small>NEW YORK · BOSTON · CHICAGO · DALLAS<br/> +ATLANTA · SAN FRANCISCO</small></p> + +<p class="front">MACMILLAN & CO., <span class="sc">Limited</span><br/> +<small>LONDON · BOMBAY · CALCUTTA<br/> +MELBOURNE</small></p> + +<p class="front">THE MACMILLAN COMPANY<br/> +OF CANADA, <span class="sc">Limited</span><br/> +<small>TORONTO</small></p> + +<h1><a class="pgnum" id="page-iii" title="- iii -"></a>THE NUMBER CONCEPT</h1> + + +<p class="front">ITS ORIGIN AND DEVELOPMENT</p> + +<p class="front sc">by<br/> +LEVI LEONARD CONANT, Ph.D.<br/> +<small>ASSOCIATE PROFESSOR OF MATHEMATICS IN THE WORCESTER<br/> +POLYTECHNIC INSTITUTE</small></p> + +<p class="front">New York<br/> +MACMILLAN AND CO.<br/> +<small>AND LONDON</small><br/> +1931</p> + +<hr class="front"/> + +<p class="front sc"><a class="pgnum" id="page-iv" title="- iv -"></a>Copyright, 1896,<br/> +By THE MACMILLAN COMPANY.</p> + +<hr class="deco"/> + +<p class="front sc">Copyright, 1924,<br/> +By EMMA B. CONANT.</p> + +<hr class="deco"/> + +<p class="front">All rights reserved—no part of this +book may be reproduced in any form +without permission in writing from +the publisher.</p> + +<hr class="deco"/> + +<p class="front">Set up and electrotyped. Published July, 1896.</p> + +<p class="front">Norwood Press<br/> +J. S. Cushing Co.—Berwick & Smith Co.<br/> +Norwood, Mass., U.S.A.</p> + + +<hr class="front"/> + + +<h2><a class="pgnum" id="page-v" title="- v -"></a>Preface.</h2> + + +<p><span class="initial">In</span> the selection of authorities which have been consulted +in the preparation of this work, and to which +reference is made in the following pages, great care has +been taken. Original sources have been drawn upon in +the majority of cases, and nearly all of these are the most +recent attainable. Whenever it has not been possible to +cite original and recent works, the author has quoted only +such as are most standard and trustworthy. In the choice +of orthography of proper names and numeral words, the +forms have, in almost all cases, been written as they were +found, with no attempt to reduce them to a systematic +English basis. In many instances this would have been +quite impossible; and, even if possible, it would have been +altogether unimportant. Hence the forms, whether German, +French, Italian, Spanish, or Danish in their transcription, +are left unchanged. Diacritical marks are omitted, +however, since the proper key could hardly be furnished +in a work of this kind.</p> + +<p>With the above exceptions, this study will, it is hoped, +be found to be quite complete; and as the subject here +investigated has never before been treated in any thorough +<a class="pgnum" id="page-vi" title="- vi -"></a>and comprehensive manner, it is hoped that this book may +be found helpful. The collections of numeral systems +illustrating the use of the binary, the quinary, and other +number systems, are, taken together, believed to be the +most extensive now existing in any language. Only the +cardinal numerals have been considered. The ordinals +present no marked peculiarities which would, in a work +of this kind, render a separate discussion necessary. Accordingly +they have, though with some reluctance, been +omitted entirely.</p> + +<p>Sincere thanks are due to those who have assisted the +author in the preparation of his materials. Especial +acknowledgment should be made to Horatio Hale, Dr. D. G. Brinton, Frank Hamilton Cushing, and Dr. A. F. +Chamberlain.</p> + + +<p class="sign"><span class="sc">Worcester, Mass.</span>, Nov. 12, 1895.</p> + + + + +<h2><a class="pgnum" id="page-vii" title="- vii -"></a>Contents.</h2> + + +<ul class="toc"> +<li class="chapter">Chapter I.</li> +<li>Counting <a href="#page-1">1</a></li> +<li class="chapter">Chapter II.</li> +<li>Number System Limits <a href="#page-21">21</a></li> +<li class="chapter">Chapter III.</li> +<li>Origin of Number Words <a href="#page-37">37</a></li> +<li class="chapter">Chapter IV.</li> +<li>Origin of Number Words (<i>continued</i>) <a href="#page-74">74</a></li> +<li class="chapter">Chapter V.</li> +<li>Miscellaneous Number Bases <a href="#page-100">100</a></li> +<li class="chapter">Chapter VI.</li> +<li>The Quinary System <a href="#page-134">134</a></li> +<li class="chapter">Chapter VII.</li> +<li>The Vigesimal System <a href="#page-176">176</a></li> +<li class="chapter"><hr/></li> +<li>Index <a href="#page-211">211</a></li> +</ul> + +<hr class="front"/> + + + +<h1><a class="pgnum" id="page-1" title="- 1 -"></a>The Number Concept: Its Origin And Development.</h1> + +<hr class="deco"/> + +<h2>Chapter I.</h2> +<h3>Counting.</h3> + + +<p><span class="initial">Among</span> the speculative questions which arise in connection +with the study of arithmetic from a historical +standpoint, the origin of number is one that has provoked +much lively discussion, and has led to a great +amount of learned research among the primitive and +savage languages of the human race. A few simple +considerations will, however, show that such research must +necessarily leave this question entirely unsettled, and will +indicate clearly that it is, from the very nature of things, +a question to which no definite and final answer can be given.</p> + +<p>Among the barbarous tribes whose languages have been +studied, even in a most cursory manner, none have ever +been discovered which did not show some familiarity with +the number concept. The knowledge thus indicated has +often proved to be most limited; not extending beyond +<a class="pgnum" id="page-2" title="- 2 -"></a>the numbers 1 and 2, or 1, 2, and 3. Examples of +this poverty of number knowledge are found among +the forest tribes of Brazil, the native races of Australia +and elsewhere, and they are considered in some +detail in the next chapter. At first thought it seems +quite inconceivable that any human being should be +destitute of the power of counting beyond 2. But +such is the case; and in a few instances languages have +been found to be absolutely destitute of pure numeral +words. The Chiquitos of Bolivia had no real numerals +whatever,<a id="FNA-1" href="#FN-1"><sup>1</sup></a> but expressed their idea for “one” by the word +<i>etama</i>, meaning alone. The Tacanas of the same country +have no numerals except those borrowed from Spanish, +or from Aymara or Peno, languages with which they have +long been in contact.<a id="FNA-2" href="#FN-2"><sup>2</sup></a> A few other South American +languages are almost equally destitute of numeral words. +But even here, rudimentary as the number sense undoubtedly +is, it is not wholly lacking; and some indirect +expression, or some form of circumlocution, shows a conception +of the difference between <i>one</i> and <i>two</i>, or at least, +between <i>one</i> and <i>many</i>.</p> + +<p>These facts must of necessity deter the mathematician +from seeking to push his investigation too far back +toward the very origin of number. Philosophers have +endeavoured to establish certain propositions concerning +this subject, but, as might have been expected, have failed +to reach any common ground of agreement. Whewell +<a class="pgnum" id="page-3" title="- 3 -"></a>has maintained that “such propositions as that two and +three make five are necessary truths, containing in them +an element of certainty beyond that which mere experience +can give.” Mill, on the other hand, argues that any +such statement merely expresses a truth derived from +early and constant experience; and in this view he is +heartily supported by Tylor.<a id="FNA-3" href="#FN-3"><sup>3</sup></a> But why this question +should provoke controversy, it is difficult for the +mathematician to understand. Either view would seem to be +correct, according to the standpoint from which the +question is approached. We know of no language in +which the suggestion of number does not appear, and we +must admit that the words which give expression to the +number sense would be among the early words to be +formed in any language. They express ideas which are, +at first, wholly concrete, which are of the greatest +possible simplicity, and which seem in many ways to be +clearly understood, even by the higher orders of the brute +creation. The origin of number would in itself, then, +appear to lie beyond the proper limits of inquiry; and the +primitive conception of number to be fundamental with +human thought.</p> + +<p>In connection with the assertion that the idea of number +seems to be understood by the higher orders of +animals, the following brief quotation from a paper by +Sir John Lubbock may not be out of place: “Leroy … +mentions a case in which a man was anxious to +shoot a crow. ‘To deceive this suspicious bird, the plan +<a class="pgnum" id="page-4" title="- 4 -"></a>was hit upon of sending two men to the watch house, +one of whom passed on, while the other remained; but +the crow counted and kept her distance. The next day +three went, and again she perceived that only two retired. +In fine, it was found necessary to send five or six men to +the watch house to put her out in her calculation. The +crow, thinking that this number of men had passed by, +lost no time in returning.’ From this he inferred that +crows could count up to four. Lichtenberg mentions a +nightingale which was said to count up to three. Every +day he gave it three mealworms, one at a time. When +it had finished one it returned for another, but after the +third it knew that the feast was over.… There is +an amusing and suggestive remark in Mr. Galton's interesting +<i>Narrative of an Explorer in Tropical South +Africa</i>. After describing the Demara's weakness in +calculations, he says: ‘Once while I watched a Demara +floundering hopelessly in a calculation on one side of +me, I observed, “Dinah,” my spaniel, equally embarrassed +on the other; she was overlooking half a dozen of her +new-born puppies, which had been removed two or three +times from her, and her anxiety was excessive, as she +tried to find out if they were all present, or if any were +still missing. She kept puzzling and running her eyes +over them backwards and forwards, but could not satisfy +herself. She evidently had a vague notion of counting, +but the figure was too large for her brain. Taking the +two as they stood, dog and Demara, the comparison +reflected no great honour on the man.…’ According to +<a class="pgnum" id="page-5" title="- 5 -"></a>my bird-nesting recollections, which I have refreshed +by more recent experience, if a nest contains four eggs, +one may safely be taken; but if two are removed, the +bird generally deserts. Here, then, it would seem as if +we had some reason for supposing that there is sufficient +intelligence to distinguish three from four. An interesting +consideration arises with reference to the number of +the victims allotted to each cell by the solitary wasps. +One species of Ammophila considers one large caterpillar +of <i>Noctua segetum</i> enough; one species of Eumenes +supplies its young with five victims; another 10, 15, and +even up to 24. The number appears to be constant in +each species. How does the insect know when her task +is fulfilled? Not by the cell being filled, for if some be +removed, she does not replace them. When she has +brought her complement she considers her task accomplished, +whether the victims are still there or not. How, +then, does she know when she has made up the number +24? Perhaps it will be said that each species feels some +mysterious and innate tendency to provide a certain number +of victims. This would, under no circumstances, be +any explanation; but it is not in accordance with the +facts. In the genus Eumenes the males are much smaller +than the females.… If the egg is male, she supplies +five; if female, 10 victims. Does she count? Certainly +this seems very like a commencement of arithmetic.”<a id="FNA-4" href="#FN-4"><sup>4</sup></a></p> + +<p>Many writers do not agree with the conclusions which +Lubbock reaches; maintaining that there is, in all such +<a class="pgnum" id="page-6" title="- 6 -"></a>instances, a perception of greater or less quantity rather +than any idea of number. But a careful consideration +of the objections offered fails entirely to weaken the +argument. Example after example of a nature similar +to those just quoted might be given, indicating on the +part of animals a perception of the difference between +1 and 2, or between 2 and 3 and 4; and any reasoning +which tends to show that it is quantity rather +than number which the animal perceives, will apply +with equal force to the Demara, the Chiquito, and +the Australian. Hence the actual origin of number +may safely be excluded from the limits of investigation, +and, for the present, be left in the field of pure +speculation.</p> + +<p>A most inviting field for research is, however, furnished +by the primitive methods of counting and of +giving visible expression to the idea of number. Our +starting-point must, of course, be the sign language, +which always precedes intelligible speech; and which +is so convenient and so expressive a method of communication +that the human family, even in its most highly +developed branches, never wholly lays it aside. It may, +indeed, be stated as a universal law, that some practical +method of numeration has, in the childhood of every +nation or tribe, preceded the formation of numeral +words.</p> + +<p>Practical methods of numeration are many in number +and diverse in kind. But the one primitive method of +counting which seems to have been almost universal +<a class="pgnum" id="page-7" title="- 7 -"></a>throughout all time is the finger method. It is a matter +of common experience and observation that every child, +when he begins to count, turns instinctively to his fingers; +and, with these convenient aids as counters, tallies +off the little number he has in mind. This method is +at once so natural and obvious that there can be no +doubt that it has always been employed by savage +tribes, since the first appearance of the human race in +remote antiquity. All research among uncivilized peoples +has tended to confirm this view, were confirmation +needed of anything so patent. Occasionally some exception +to this rule is found; or some variation, such as is +presented by the forest tribes of Brazil, who, instead of +counting on the fingers themselves, count on the joints +of their fingers.<a id="FNA-5" href="#FN-5"><sup>5</sup></a> As the entire number system of these +tribes appears to be limited to <i>three</i>, this variation is +no cause for surprise.</p> + +<p>The variety in practical methods of numeration observed +among savage races, and among civilized peoples +as well, is so great that any detailed account of them +would be almost impossible. In one region we find +sticks or splints used; in another, pebbles or shells; in +another, simple scratches, or notches cut in a stick, +Robinson Crusoe fashion; in another, kernels or little +heaps of grain; in another, knots on a string; and so +on, in diversity of method almost endless. Such are the +devices which have been, and still are, to be found in +<a class="pgnum" id="page-8" title="- 8 -"></a>the daily habit of great numbers of Indian, negro, +Mongolian, and Malay tribes; while, to pass at a single +step to the other extremity of intellectual development, +the German student keeps his beer score by +chalk marks on the table or on the wall. But back of +all these devices, and forming a common origin to which +all may be referred, is the universal finger method; the +method with which all begin, and which all find too +convenient ever to relinquish entirely, even though +their civilization be of the highest type. Any such +mode of counting, whether involving the use of the +fingers or not, is to be regarded simply as an extraneous +aid in the expression or comprehension of an idea which +the mind cannot grasp, or cannot retain, without assistance. +The German student scores his reckoning with +chalk marks because he might otherwise forget; while +the Andaman Islander counts on his fingers because he +has no other method of counting,—or, in other words, +of grasping the idea of number. A single illustration +may be given which typifies all practical methods of +numeration. More than a century ago travellers in +Madagascar observed a curious but simple mode of +ascertaining the number of soldiers in an army.<a id="FNA-6" href="#FN-6"><sup>6</sup></a> Each +soldier was made to go through a passage in the presence +of the principal chiefs; and as he went through, +a pebble was dropped on the ground. This continued +until a heap of 10 was obtained, when one was set aside +<a class="pgnum" id="page-9" title="- 9 -"></a>and a new heap begun. Upon the completion of 10 +heaps, a pebble was set aside to indicate 100; and so +on until the entire army had been numbered. Another +illustration, taken from the very antipodes of Madagascar, +recently found its way into print in an incidental +manner,<a id="FNA-7" href="#FN-7"><sup>7</sup></a> and is so good that it deserves a place +beside de Flacourt's time-honoured example. Mom Cely, +a Southern negro of unknown age, finds herself in debt +to the storekeeper; and, unwilling to believe that the +amount is as great as he represents, she proceeds to +investigate the matter in her own peculiar way. She +had “kept a tally of these purchases by means of a +string, in which she tied commemorative knots.” When +her creditor “undertook to make the matter clear to +Cely's comprehension, he had to proceed upon a system +of her own devising. A small notch was cut in a smooth +white stick for every dime she owed, and a large notch +when the dimes amounted to a dollar; for every five +dollars a string was tied in the fifth big notch, Cely +keeping tally by the knots in her bit of twine; thus, +when two strings were tied about the stick, the ten dollars +were seen to be an indisputable fact.” This interesting +method of computing the amount of her debt, +whether an invention of her own or a survival of the +African life of her parents, served the old negro woman's +purpose perfectly; and it illustrates, as well as a score +of examples could, the methods of numeration to which +the children of barbarism resort when any number is +<a class="pgnum" id="page-10" title="- 10 -"></a>to be expressed which exceeds the number of counters +with which nature has provided them. The fingers are, +however, often employed in counting numbers far above +the first decade. After giving the Il-Oigob numerals up +to 60, Müller adds:<a id="FNA-8" href="#FN-8"><sup>8</sup></a> “Above 60 all numbers, indicated +by the proper figure pantomime, are expressed by means +of the word <i>ipi</i>.” We know, moreover, that many of the +American Indian tribes count one ten after another on +their fingers; so that, whatever number they are endeavouring +to indicate, we need feel no surprise if the savage +continues to use his fingers throughout the entire extent +of his counts. In rare instances we find tribes which, like +the Mairassis of the interior of New Guinea, appear to +use nothing but finger pantomime.<a id="FNA-9" href="#FN-9"><sup>9</sup></a> This tribe, though +by no means destitute of the number sense, is said to +have no numerals whatever, but to use the single word +<i>awari</i> with each show of fingers, no matter how few or +how many are displayed.</p> + +<p>In the methods of finger counting employed by savages +a considerable degree of uniformity has been observed. +Not only does he use his fingers to assist him +in his tally, but he almost always begins with the little +finger of his left hand, thence proceeding towards +the thumb, which is 5. From this point onward the +method varies. Sometimes the second 5 also is told off +on the left hand, the same order being observed as in +the first 5; but oftener the fingers of the right hand +<a class="pgnum" id="page-11" title="- 11 -"></a>are used, with a reversal of the order previously employed; +<i>i.e.</i> the thumb denotes 6, the index finger 7, +and so on to the little finger, which completes the +count to 10.</p> + +<p>At first thought there would seem to be no good +reason for any marked uniformity of method in finger +counting. Observation among children fails to detect +any such thing; the child beginning, with almost entire +indifference, on the thumb or on the little finger of the +left hand. My own observation leads to the conclusion +that very young children have a slight, though not +decided preference for beginning with the thumb. +Experiments in five different primary rooms in the public +schools of Worcester, Mass., showed that out of a +total of 206 children, 57 began with the little finger +and 149 with the thumb. But the fact that nearly +three-fourths of the children began with the thumb, +and but one-fourth with the little finger, is really far +less significant than would appear at first thought. +Children of this age, four to eight years, will count in +either way, and sometimes seem at a loss themselves +to know where to begin. In one school room where +this experiment was tried the teacher incautiously asked +one child to count on his fingers, while all the other +children in the room watched eagerly to see what he +would do. He began with the little finger—and so did +every child in the room after him. In another case +the same error was made by the teacher, and the child +first asked began with the thumb. Every other child +<a class="pgnum" id="page-12" title="- 12 -"></a>in the room did the same, each following, consciously +or unconsciously, the example of the leader. The results +from these two schools were of course rejected +from the totals which are given above; but they serve +an excellent purpose in showing how slight is the preference +which very young children have in this particular. +So slight is it that no definite law can be +postulated of this age; but the tendency seems to be +to hold the palm of the hand downward, and then +begin with the thumb. The writer once saw a boy +about seven years old trying to multiply 3 by 6; and +his method of procedure was as follows: holding his +left hand with its palm down, he touched with the +forefinger of his right hand the thumb, forefinger, and +middle finger successively of his left hand. Then returning +to his starting-point, he told off a second three +in the same manner. This process he continued until +he had obtained 6 threes, and then he announced his +result correctly. If he had been a few years older, he +might not have turned so readily to his thumb as a +starting-point for any digital count. The indifference +manifested by very young children gradually disappears, +and at the age of twelve or thirteen the tendency is +decidedly in the direction of beginning with the little +finger. Fully three-fourths of all persons above that +age will be found to count from the little finger toward +the thumb, thus reversing the proportion that was found +to obtain in the primary school rooms examined.</p> + +<p>With respect to finger counting among civilized +<a class="pgnum" id="page-13" title="- 13 -"></a>peoples, we fail, then, to find any universal law; the +most that can be said is that more begin with the little +finger than with the thumb. But when we proceed to +the study of this slight but important particular among +savages, we find them employing a certain order of +succession with such substantial uniformity that the +conclusion is inevitable that there must lie back of this +some well-defined reason, or perhaps instinct, which +guides them in their choice. This instinct is undoubtedly +the outgrowth of the almost universal right-handedness +of the human race. In finger counting, whether +among children or adults, the beginning is made on +the left hand, except in the case of left-handed individuals; +and even then the start is almost as likely to +be on the left hand as on the right. Savage tribes, as +might be expected, begin with the left hand. Not +only is this custom almost invariable, when tribes as +a whole are considered, but the little finger is nearly +always called into requisition first. To account for this +uniformity, Lieutenant Gushing gives the following +theory,<a id="FNA-10" href="#FN-10"><sup>10</sup></a> which is well considered, and is based on the +results of careful study and observation among the Zuñi +Indians of the Southwest: “Primitive man when abroad +never lightly quit hold of his weapons. If he wanted to +count, he did as the Zuñi afield does to-day; he tucked +his instrument under his left arm, thus constraining the +latter, but leaving the right hand free, that he might +check off with it the fingers of the rigidly elevated left +<a class="pgnum" id="page-14" title="- 14 -"></a>hand. From the nature of this position, however, the +palm of the left hand was presented to the face of the +counter, so that he had to begin his score on the little +finger of it, and continue his counting from the right +leftward. An inheritance of this may be detected to-day +in the confirmed habit the Zuñi has of gesticulating +from the right leftward, with the fingers of the +right hand over those of the left, whether he be counting +and summing up, or relating in any orderly manner.” +Here, then, is the reason for this otherwise unaccountable +phenomenon. If savage man is universally right-handed, +he will almost inevitably use the index finger of his right +hand to mark the fingers counted, and he will begin his +count just where it is most convenient. In his case it +is with the little finger of the left hand. In the case +of the child trying to multiply 3 by 6, it was with the +thumb of the same hand. He had nothing to tuck under +his arm; so, in raising his left hand to a position where +both eye and counting finger could readily run over its +fingers, he held the palm turned away from his face. +The same choice of starting-point then followed as with +the savage—the finger nearest his right hand; only in +this case the finger was a thumb. The deaf mute is +sometimes taught in this manner, which is for him an +entirely natural manner. A left-handed child might +be expected to count in a left-to-right manner, beginning, +probably, with the thumb of his right hand.</p> + +<p>To the law just given, that savages begin to count +on the little finger of the left hand, there have been +<a class="pgnum" id="page-15" title="- 15 -"></a>a few exceptions noted; and it has been observed that +the method of progression on the second hand is by no +means as invariable as on the first. The Otomacs<a id="FNA-11" href="#FN-11"><sup>11</sup></a> of +South America began their count with the thumb, and +to express the number 3 would use the thumb, forefinger, +and middle finger. The Maipures,<a id="FNA-12" href="#FN-12"><sup>12</sup></a> oddly enough, +seem to have begun, in some cases at least, with the +forefinger; for they are reported as expressing 3 by +means of the fore, middle, and ring fingers. The Andamans<a id="FNA-13" href="#FN-13"><sup>13</sup></a> +begin with the little finger of either hand, tapping +the nose with each finger in succession. If they have +but one to express, they use the forefinger of either hand, +pronouncing at the same time the proper word. The +Bahnars,<a id="FNA-14" href="#FN-14"><sup>14</sup></a> one of the native tribes of the interior of +Cochin China, exhibit no particular order in the sequence +of fingers used, though they employ their digits freely +to assist them in counting. Among certain of the negro +tribes of South Africa<a id="FNA-15" href="#FN-15"><sup>15</sup></a> the little finger of the right hand +is used for 1, and their count proceeds from right to left. +With them, 6 is the thumb of the left hand, 7 the forefinger, +and so on. They hold the palm downward instead +of upward, and thus form a complete and striking exception +to the law which has been found to obtain with such +<a class="pgnum" id="page-16" title="- 16 -"></a>substantial uniformity in other parts of the uncivilized +world. In Melanesia a few examples of preference for +beginning with the thumb may also be noticed. In the +Banks Islands the natives begin by turning down the +thumb of the right hand, and then the fingers in succession +to the little finger, which is 5. This is followed +by the fingers of the left hand, both hands with closed +fists being held up to show the completed 10. In Lepers' +Island, they begin with the thumb, but, having reached +5 with the little finger, they do not pass to the other +hand, but throw up the fingers they have turned down, +beginning with the forefinger and keeping the thumb +for 10.<a id="FNA-16" href="#FN-16"><sup>16</sup></a> In the use of the single hand this people is +quite peculiar. The second 5 is almost invariably told +off by savage tribes on the second hand, though in +passing from the one to the other primitive man does +not follow any invariable law. He marks 6 with either +the thumb or the little finger. Probably the former is +the more common practice, but the statement cannot be +made with any degree of certainty. Among the Zulus +the sequence is from thumb to thumb, as is the case +among the other South African tribes just mentioned; +while the Veis and numerous other African tribes pass +from thumb to little finger. The Eskimo, and nearly +all the American Indian tribes, use the correspondence +between 6 and the thumb; but this habit is by no means +universal. Respecting progression from right to left or +<a class="pgnum" id="page-17" title="- 17 -"></a>left to right on the toes, there is no general law with +which the author is familiar. Many tribes never use +the toes in counting, but signify the close of the first 10 +by clapping the hands together, by a wave of the right +hand, or by designating some object; after which the +fingers are again used as before.</p> + +<p>One other detail in finger counting is worthy of a +moment's notice. It seems to have been the opinion +of earlier investigators that in his passage from one +finger to the next, the savage would invariably bend +down, or close, the last finger used; that is, that the +count began with the fingers open and outspread. This +opinion is, however, erroneous. Several of the Indian +tribes of the West<a id="FNA-17" href="#FN-17"><sup>17</sup></a> begin with the hand clenched, and +open the fingers one by one as they proceed. This +method is much less common than the other, but that +it exists is beyond question.</p> + +<p>In the Muralug Island, in the western part of Torres +Strait, a somewhat remarkable method of counting formerly +existed, which grew out of, and is to be regarded +as an extension of, the digital method. Beginning with +the little finger of the left hand, the natives counted +up to 5 in the usual manner, and then, instead of +passing to the other hand, or repeating the count on +the same fingers, they expressed the numbers from 6 +to 10 by touching and naming successively the left +wrist, left elbow, left shoulder, left breast, and sternum. +Then the numbers from 11 to 19 were indicated by +<a class="pgnum" id="page-18" title="- 18 -"></a>the use, in inverse order, of the corresponding portions +of the right side, arm, and hand, the little finger of +the right hand signifying 19. The words used were +in each case the actual names of the parts touched; +the same word, for example, standing for 6 and 14; +but they were never used in the numerical sense +unless accompanied by the proper gesture, and bear no +resemblance to the common numerals, which are but +few in number. This method of counting is rapidly +dying out among the natives of the island, and is at +the present time used only by old people.<a id="FNA-18" href="#FN-18"><sup>18</sup></a> Variations +on this most unusual custom have been found to exist +in others of the neighbouring islands, but none were +exactly similar to it. One is also reminded by it of +a custom<a id="FNA-19" href="#FN-19"><sup>19</sup></a> which has for centuries prevailed among bargainers +in the East, of signifying numbers by touching +the joints of each other's fingers under a cloth. Every +joint has a special signification; and the entire system +is undoubtedly a development from finger counting. +The buyer or seller will by this method express 6 +or 60 by stretching out the thumb and little finger +and closing the rest of the fingers. The addition of +the fourth finger to the two thus used signifies 7 +or 70; and so on. <del title="Probably spurious quote as there is no closing quote following nor a source mentioned.">“</del>It is said that between two brokers +settling a price by thus snipping with the fingers, +cleverness in bargaining, offering a little more, hesitating, +<a class="pgnum" id="page-19" title="- 19 -"></a>expressing an obstinate refusal to go further, etc., +are as clearly indicated as though the bargaining were +being carried on in words.</p> + +<p>The place occupied, in the intellectual development +of man, by finger counting and by the many other artificial +methods of reckoning,—pebbles, shells, knots, the +abacus, etc.,—seems to be this: The abstract processes +of addition, subtraction, multiplication, division, and even +counting itself, present to the mind a certain degree of +difficulty. To assist in overcoming that difficulty, these +artificial aids are called in; and, among savages of a low +degree of development, like the Australians, they make +counting possible. A little higher in the intellectual +scale, among the American Indians, for example, they +are employed merely as an artificial aid to what could +be done by mental effort alone. Finally, among semi-civilized +and civilized peoples, the same processes are +retained, and form a part of the daily life of almost +every person who has to do with counting, reckoning, +or keeping tally in any manner whatever. They are +no longer necessary, but they are so convenient and +so useful that civilization can never dispense with them. +The use of the abacus, in the form of the ordinary +numeral frame, has increased greatly within the past +few years; and the time may come when the abacus in +its proper form will again find in civilized countries a +use as common as that of five centuries ago.</p> + +<p>In the elaborate calculating machines of the present, +such as are used by life insurance actuaries and others +<a class="pgnum" id="page-20" title="- 20 -"></a>having difficult computations to make, we have the extreme +of development in the direction of artificial aid +to reckoning. But instead of appearing merely as an +extraneous aid to a defective intelligence, it now presents +itself as a machine so complex that a high degree +of intellectual power is required for the mere grasp +of its construction and method of working.</p> + + + + +<h2><a class="pgnum" id="page-21" title="- 21 -"></a>Chapter II.</h2> +<h3>Number System Limits.</h3> + + +<p><span class="initial">With</span> respect to the limits to which the number +systems of the various uncivilized races of the earth +extend, recent anthropological research has developed +many interesting facts. In the case of the Chiquitos +and a few other native races of Bolivia we found no +distinct number sense at all, as far as could be judged +from the absence, in their language, of numerals in the +proper sense of the word. How they indicated any +number greater than <i>one</i> is a point still requiring +investigation. In all other known instances we find +actual number systems, or what may for the sake of +uniformity be dignified by that name. In many cases, +however, the numerals existing are so few, and the +ability to count is so limited, that the term <i>number +system</i> is really an entire misnomer.</p> + +<p>Among the rudest tribes, those whose mode of living +approaches most nearly to utter savagery, we find a +certain uniformity of method. The entire number +system may consist of but two words, <i>one</i> and <i>many</i>; +or of three words, <i>one</i>, <i>two</i>, <i>many</i>. Or, the count may +proceed to 3, 4, 5, 10, 20, or 100; passing always, +or almost always, from the distinct numeral limit +<a class="pgnum" id="page-22" title="- 22 -"></a>to the indefinite <i>many</i> or several, which serves for the +expression of any number not readily grasped by the +mind. As a matter of fact, most races count as high +as 10; but to this statement the exceptions are so +numerous that they deserve examination in some detail. +In certain parts of the world, notably among the +native races of South America, Australia, and many +of the islands of Polynesia and Melanesia, a surprising +paucity of numeral words has been observed. The Encabellada +of the Rio Napo have but two distinct numerals; +<i>tey</i>, 1, and <i>cayapa</i>, 2.<a id="FNA-20" href="#FN-20"><sup>20</sup></a> The Chaco languages<a id="FNA-21" href="#FN-21"><sup>21</sup></a> of +the Guaycuru stock are also notably poor in this respect. +In the Mbocobi dialect of this language the +only native numerals are <i>yña tvak</i>, 1, and <i>yfioaca</i>, 2. +The Puris<a id="FNA-22" href="#FN-22"><sup>22</sup></a> count <i>omi</i>, 1, <i>curiri</i>, 2, <i>prica</i>, many; and +the Botocudos<a id="FNA-23" href="#FN-23"><sup>23</sup></a> <i>mokenam</i>, 1, <i>uruhu</i>, many. The Fuegans,<a id="FNA-24" href="#FN-24"><sup>24</sup></a> +supposed to have been able at one time to count +to 10, have but three numerals,—<i>kaoueli</i>, 1, <i>compaipi</i>, 2, +<i>maten</i>, 3. The Campas of Peru<a id="FNA-25" href="#FN-25"><sup>25</sup></a> possess only three +separate words for the expression of number,—<i>patrio</i>, +1, <i>pitteni</i>, 2, <i>mahuani</i>, 3. Above 3 they proceed by +combinations, as 1 and 3 for 4, 1 and 1 and 3 for 5. +Counting above 10 is, however, entirely inconceivable +to them, and any number beyond that limit they indicate +<a class="pgnum" id="page-23" title="- 23 -"></a>by <i>tohaine</i>, many. The Conibos,<a id="FNA-26" href="#FN-26"><sup>26</sup></a> of the same +region, had, before their contact with the Spanish, only +<i>atchoupre</i>, 1, and <i>rrabui</i>, 2; though they made some +slight progress above 2 by means of reduplication. +The Orejones, one of the low, degraded tribes of the +Upper Amazon,<a id="FNA-27" href="#FN-27"><sup>27</sup></a> have no names for number except +<i>nayhay</i>, 1, <i>nenacome</i>, 2, <i>feninichacome</i>, 3, <i>ononoeomere</i>, 4. +In the extensive vocabularies given by Von Martins,<a id="FNA-28" href="#FN-28"><sup>28</sup></a> +many similar examples are found. For the Bororos he +gives only <i>couai</i>, 1, <i>maeouai</i>, 2, <i>ouai</i>, 3. The last word, +with the proper finger pantomime, serves also for any +higher number which falls within the grasp of their comprehension. +The Guachi manage to reach 5, but their +numeration is of the rudest kind, as the following +scale shows: <i>tamak</i>, 1, <i>eu-echo,</i> 2, <i>eu-echo-kailau,</i> 3, <i>eu-echo-way,</i> +4, <i>localau</i>, 5. The Carajas counted by a +scale equally rude, and their conception of number +seemed equally vague, until contact with the neighbouring +tribes furnished them with the means of going +beyond their original limit. Their scale shows clearly +the uncertain, feeble number sense which is so marked +in the interior of South America. It contains <i>wadewo</i>, +1, <i>wadebothoa</i>, 2, <i>wadeboaheodo</i>, 3, <i>wadebojeodo</i>, 4, <i>wadewajouclay</i>, +5, <i>wadewasori</i>, 6, or many.</p> + +<p><a class="pgnum" id="page-24" title="- 24 -"></a>Turning to the languages of the extinct, or fast vanishing, +tribes of Australia, we find a still more noteworthy +absence of numeral expressions. In the Gudang +dialect<a id="FNA-29" href="#FN-29"><sup>29</sup></a> but two numerals are found—<i>pirman</i>, 1, and +<i>ilabiu</i>, 2; in the Weedookarry, <i>ekkamurda</i>, 1, and <i>kootera</i>, +2; and in the Queanbeyan, <i>midjemban</i>, 1, and <i>bollan</i>, +2. In a score or more of instances the numerals stop +at 3. The natives of Keppel Bay count <i>webben</i>, 1, <i>booli</i>, +2, <i>koorel</i>, 3; of the Boyne River, <i>karroon</i>, 1, <i>boodla</i>, 2, +<i>numma</i>, 3; of the Flinders River, <i>kooroin</i>, 1, <i>kurto</i>, 2, +<i>kurto kooroin</i>, 3; at the mouth of the Norman River, +<i>lum</i>, 1, <i>buggar</i>, 2, <i>orinch</i>, 3; the Eaw tribe, <i>koothea</i>, 1, +<i>woother</i>, 2, <i>marronoo</i>, 3; the Moree, <i>mal</i>, 1, <i>boolar</i>, +2, <i>kooliba</i>, 3; the Port Essington,<a id="FNA-30" href="#FN-30"><sup>30</sup></a> <i>erad</i>, 1, <i>nargarick</i>, +2, <i>nargarickelerad</i>, 3; the Darnly Islanders,<a id="FNA-31" href="#FN-31"><sup>31</sup></a> <i>netat</i>, 1, +<i>naes</i>, 2, <i>naesa netat</i>, 3; and so on through a long list +of tribes whose numeral scales are equally scanty. A +still larger number of tribes show an ability to count +one step further, to 4; but beyond this limit the majority +of Australian and Tasmanian tribes do not go. It +seems most remarkable that any human being should +possess the ability to count to 4, and not to 5. The +number of fingers on one hand furnishes so obvious a +limit to any of these rudimentary systems, that positive +evidence is needed before one can accept the statement. +<a class="pgnum" id="page-25" title="- 25 -"></a>A careful examination of the numerals in upwards of a +hundred Australian dialects leaves no doubt, however, +that such is the fact. The Australians in almost all cases +count by pairs; and so pronounced is this tendency that +they pay but little attention to the fingers. Some tribes +do not appear ever to count beyond 2—a single pair. +Many more go one step further; but if they do, they are +as likely as not to designate their next numeral as two-one, +or possibly, one-two. If this step is taken, we may +or may not find one more added to it, thus completing +the second pair. Still, the Australian's capacity for +understanding anything which pertains to number is so +painfully limited that even here there is sometimes an +indefinite expression formed, as many, heap, or plenty, +instead of any distinct numeral; and it is probably true +that no Australian language contains a pure, simple +numeral for 4. Curr, the best authority on this subject, +believes that, where a distinct word for 4 is given, +investigators have been deceived in every case.<a id="FNA-32" href="#FN-32"><sup>32</sup></a> If +counting is carried beyond 4, it is always by means of +reduplication. A few tribes gave expressions for 5, +fewer still for 6, and a very small number appeared +able to reach 7. Possibly the ability to count extended +still further; but if so, it consisted undoubtedly in +reckoning one pair after another, without any consciousness +whatever of the sum total save as a larger +number.</p> + +<p>The numerals of a few additional tribes will show +<a class="pgnum" id="page-26" title="- 26 -"></a>clearly that all distinct perception of number is lost as +soon as these races attempt to count above 3, or at most, +4. The Yuckaburra<a id="FNA-33" href="#FN-33"><sup>33</sup></a> natives can go no further than +<i>wigsin</i>, 1, <i>bullaroo</i>, 2, <i>goolbora</i>, 3. Above here all is +referred to as <i>moorgha</i>, many. The Marachowies<a id="FNA-34" href="#FN-34"><sup>34</sup></a> have +but three distinct numerals,—<i>cooma</i>, 1, <i>cootera</i>, 2, <i>murra</i>, +3. For 4 they say <i>minna</i>, many. At Streaky Bay we +find a similar list, with the same words, <i>kooma</i> and +<i>kootera</i>, for 1 and 2, but entirely different terms, <i>karboo</i> +and <i>yalkata</i> for 3 and many. The same method obtains +in the Minnal Yungar tribe, where the only numerals +are <i>kain</i>, 1, <i>kujal</i>, 2, <i>moa</i>, 3, and <i>bulla</i>, plenty. In the +Pinjarra dialect we find <i>doombart</i>, 1, <i>gugal</i>, 2, <i>murdine</i>, +3, <i>boola</i>, plenty; and in the dialect described as belonging +to “Eyre's Sand Patch,” three definite terms are +given—<i>kean</i>, 1, <i>koojal</i>, 2, <i>yalgatta</i>, 3, while a fourth, +<i>murna</i>, served to describe anything greater. In all +these examples the fourth numeral is indefinite; and +the same statement is true of many other Australian +languages. But more commonly still we find 4, and +perhaps 3 also, expressed by reduplication. In the Port +Mackay dialect<a id="FNA-35" href="#FN-35"><sup>35</sup></a> the latter numeral is compound, the +count being <i>warpur</i>, 1, <i>boolera</i>, 2, <i>boolera warpur</i>, 3. For +4 the term is not given. In the dialect which prevailed +between the Albert and Tweed rivers<a id="FNA-36" href="#FN-36"><sup>36</sup></a> the scale appears +<a class="pgnum" id="page-27" title="- 27 -"></a>as <i>yaburu</i>, 1, <i>boolaroo</i>, 2, <i>boolaroo yaburu</i>, 3, and <i>gurul</i> for +4 or anything beyond. The Wiraduroi<a id="FNA-37" href="#FN-37"><sup>37</sup></a> have <i>numbai</i>, 1, +<i>bula</i>, 2, <i>bula numbai</i>, 3, <i>bungu</i>, 4, or many, and <i>bungu galan</i> +or <i>bian galan</i>, 5, or very many. The Kamilaroi<a id="FNA-38" href="#FN-38"><sup>38</sup></a> scale +is still more irregular, compounding above 4 with little +apparent method. The numerals are <i>mal</i>, 1, <i>bular</i>, 2, +<i>guliba</i>, 3, <i>bular bular</i>, 4, <i>bular guliba</i>, 5, <i>guliba guliba</i>, 6. +The last two numerals show that 5 is to these natives +simply 2-3, and 6 is 3-3. For additional examples of a +similar nature the extended list of Australian scales +given in Chapter V. may be consulted.</p> + +<p>Taken as a whole, the Australian and Tasmanian tribes +seem to have been distinctly inferior to those of South +America in their ability to use and to comprehend +numerals. In all but two or three cases the Tasmanians<a id="FNA-39" href="#FN-39"><sup>39</sup></a> +were found to be unable to proceed beyond 2; and as the +foregoing examples have indicated, their Australian +neighbours were but little better off. In one or two instances +we do find Australian numeral scales which reach +10, and perhaps we may safely say 20. One of these is +given in full in a subsequent chapter, and its structure +gives rise to the suspicion that it was originally as limited +as those of kindred tribes, and that it underwent a considerable +development after the natives had come in contact +with the Europeans. There is good reason to believe +that no Australian in his wild state could ever count +intelligently to 7.<a id="FNA-40" href="#FN-40"><sup>40</sup></a></p> + + +<p><a class="pgnum" id="page-28" title="- 28 -"></a>In certain portions of Asia, Africa, Melanesia, Polynesia, +and North America, are to be found races whose +number systems are almost and sometimes quite as limited +as are those of the South. American and Australian +tribes already cited, but nowhere else do we find these +so abundant as in the two continents just mentioned, +where example after example might be cited of tribes +whose ability to count is circumscribed within the narrowest +limits. The Veddas<a id="FNA-41" href="#FN-41"><sup>41</sup></a> of Ceylon have but two +numerals, <i>ekkameī</i>, 1, <i>dekkameï</i>, 2. Beyond this they +count <i>otameekaï, otameekaï, otameekaï</i>, etc.; <i>i.e.</i> “and +one more, and one more, and one more,” and so on indefinitely. +The Andamans,<a id="FNA-42" href="#FN-42"><sup>42</sup></a> inhabitants of a group of +islands in the Bay of Bengal, are equally limited in +their power of counting. They have <i>ubatulda</i>, 1, and +<i>ikporda</i>, 2; but they can go no further, except in a +manner similar to that of the Veddas. Above two they +proceed wholly by means of the fingers, saying as they +tap the nose with each successive finger, <i>anka</i>, “and +this.” Only the more intelligent of the Andamans can +count at all, many of them seeming to be as nearly destitute +of the number sense as it is possible for a human +being to be. The Bushmen<a id="FNA-43" href="#FN-43"><sup>43</sup></a> of South Africa have but +two numerals, the pronunciation of which can hardly be +indicated without other resources than those of the English +alphabet. Their word for 3 means, simply, many, +<a class="pgnum" id="page-29" title="- 29 -"></a>as in the case of some of the Australian tribes. The +Watchandies<a id="FNA-44" href="#FN-44"><sup>44</sup></a> have but two simple numerals, and their +entire number system is <i>cooteon</i>, 1, <i>utaura</i>, 2, <i>utarra +cooteoo</i>, 3, <i>atarra utarra</i>, 4. Beyond this they can only +say, <i>booltha</i>, many, and <i>booltha bat</i>, very many. Although +they have the expressions here given for 3 and 4, they are +reluctant to use them, and only do so when absolutely +required. The natives of Lower California<a id="FNA-45" href="#FN-45"><sup>45</sup></a> cannot count +above 5. A few of the more intelligent among them +understand the meaning of 2 fives, but this number +seems entirely beyond the comprehension of the ordinary +native. The Comanches, curiously enough, are so reluctant +to employ their number words that they appear to +prefer finger pantomime instead, thus giving rise to the +impression which at one time became current, that they +had no numerals at all for ordinary counting.</p> + +<p>Aside from the specific examples already given, a considerable +number of sweeping generalizations may be +made, tending to show how rudimentary the number +sense may be in aboriginal life. Scores of the native +dialects of Australia and South America have been found +containing number systems but little more extensive than +those alluded to above. The negro tribes of Africa give +the same testimony, as do many of the native races of +Central America, Mexico, and the Pacific coast of the +United States and Canada, the northern part of Siberia, +Greenland, Labrador, and the arctic archipelago. In +<a class="pgnum" id="page-30" title="- 30 -"></a>speaking of the Eskimos of Point Barrow, Murdoch<a id="FNA-46" href="#FN-46"><sup>46</sup></a> +says: “It was not easy to obtain any accurate information +about the numeral system of these people, since in +ordinary conversation they are not in the habit of specifying +any numbers above five.” Counting is often carried +higher than this among certain of these northern +tribes, but, save for occasional examples, it is limited at +best. Dr. Franz Boas, who has travelled extensively +among the Eskimos, and whose observations are always +of the most accurate nature, once told the author that +he never met an Eskimo who could count above 15. +Their numerals actually do extend much higher; and +a stray numeral of Danish origin is now and then met +with, showing that the more intelligent among them are +able to comprehend numbers of much greater magnitude +than this. But as Dr. Boas was engaged in active +work among them for three years, we may conclude +that the Eskimo has an arithmetic but little more extended +than that which sufficed for the Australians and +the forest tribes of Brazil. Early Russian explorers +among the northern tribes of Siberia noticed the same +difficulty in ordinary, every-day reckoning among the +natives. At first thought we might, then, state it as +a general law that those races which are lowest in the +scale of civilization, have the feeblest number sense also; +or in other words, the least possible power of grasping +the abstract idea of number.</p> + + +<p><a class="pgnum" id="page-31" title="- 31 -"></a>But to this law there are many and important exceptions. +The concurrent testimony of explorers seems to +be that savage races possess, in the great majority of +cases, the ability to count at least as high as 10. This +limit is often extended to 20, and not infrequently to +100. Again, we find 1000 as the limit; or perhaps +10,000; and sometimes the savage carries his number +system on into the hundreds of thousands or millions. +Indeed, the high limit to which some savage races +carry their numeration is far more worthy of remark +than the entire absence of the number sense exhibited +by others of apparently equal intelligence. If the life +of any tribe is such as to induce trade and barter with +their neighbours, a considerable quickness in reckoning +will be developed among them. Otherwise this power +will remain dormant because there is but little in the +ordinary life of primitive man to call for its exercise.</p> + +<p>In giving 1, 2, 3, 5, 10, or any other small number +as a system limit, it must not be overlooked that this +limit mentioned is in all cases the limit of the spoken +numerals at the savage's command. The actual ability +to count is almost always, and one is tempted to say +always, somewhat greater than their vocabularies would +indicate. The Bushman has no number word that +will express for him anything higher than 2; but +with the assistance of his fingers he gropes his way on +as far as 10. The Veddas, the Andamans, the Guachi, +the Botocudos, the Eskimos, and the thousand and one +other tribes which furnish such scanty numeral systems, +<a class="pgnum" id="page-32" title="- 32 -"></a>almost all proceed with more or less readiness as +far as their fingers will carry them. As a matter of +fact, this limit is frequently extended to 20; the toes, +the fingers of a second man, or a recount of the savage's +own fingers, serving as a tale for the second +10. Allusion is again made to this in a later chapter, +where the subject of counting on the fingers and toes +is examined more in detail.</p> + +<p>In saying that a savage can count to 10, to 20, or to +100, but little idea is given of his real mental conception +of any except the smallest numbers. Want of +familiarity with the use of numbers, and lack of convenient +means of comparison, must result in extreme +indefiniteness of mental conception and almost entire +absence of exactness. The experience of Captain +Parry,<a id="FNA-47" href="#FN-47"><sup>47</sup></a> who found that the Eskimos made mistakes +before they reached 7, and of Humboldt,<a id="FNA-48" href="#FN-48"><sup>48</sup></a> who says +that a Chayma might be made to say that his age +was either 18 or 60, has been duplicated by all +investigators who have had actual experience among +savage races. Nor, on the other hand, is the development +of a numeral system an infallible index of mental +power, or of any real approach toward civilization. A +continued use of the trading and bargaining faculties +must and does result in a familiarity with numbers +sufficient to enable savages to perform unexpected feats +in reckoning. Among some of the West African tribes +<a class="pgnum" id="page-33" title="- 33 -"></a>this has actually been found to be the case; and among +the Yorubas of Abeokuta<a id="FNA-49" href="#FN-49"><sup>49</sup></a> the extraordinary saying, +“You may seem very clever, but you can't tell nine +times nine,” shows how surprisingly this faculty has +been developed, considering the general condition of +savagery in which the tribe lived. There can be no +doubt that, in general, the growth of the number sense +keeps pace with the growth of the intelligence in other +respects. But when it is remembered that the Tonga +Islanders have numerals up to 100,000, and the Tembus, +the Fingoes, the Pondos, and a dozen other South +African tribes go as high as 1,000,000; and that Leigh +Hunt never could learn the multiplication table, one +must confess that this law occasionally presents to our +consideration remarkable exceptions.</p> + +<p>While considering the extent of the savage's arithmetical +knowledge, of his ability to count and to grasp the +meaning of number, it may not be amiss to ask ourselves +the question, what is the extent of the development of +our own number sense? To what limit can we absorb +the idea of number, with a complete appreciation of the +idea of the number of units involved in any written or +spoken quantity? Our perfect system of numeration +enables us to express without difficulty any desired number, +no matter how great or how small it be. But how +much of actually clear comprehension does the number +thus expressed convey to the mind? We say that one +place is 100 miles from another; that A paid B 1000 +<a class="pgnum" id="page-34" title="- 34 -"></a>dollars for a certain piece of property; that a given +city contains 10,000 inhabitants; that 100,000 bushels +of wheat were shipped from Duluth or Odessa on such +a day; that 1,000,000 feet of lumber were destroyed by +the fire of yesterday,—and as we pass from the smallest +to the largest of the numbers thus instanced, and from +the largest on to those still larger, we repeat the question +just asked; and we repeat it with a new sense of our +own mental limitation. The number 100 unquestionably +stands for a distinct conception. Perhaps the same +may be said for 1000, though this could not be postulated +with equal certainty. But what of 10,000? If that +number of persons were gathered together into a single +hall or amphitheatre, could an estimate be made by the +average onlooker which would approximate with any +degree of accuracy the size of the assembly? Or if an +observer were stationed at a certain point, and 10,000 +persons were to pass him in single file without his counting +them as they passed, what sort of an estimate would +he make of their number? The truth seems to be that +our mental conception of number is much more limited +than is commonly thought, and that we unconsciously +adopt some new unit as a standard of comparison when +we wish to render intelligible to our minds any number +of considerable magnitude. For example, we say that +A has a fortune of $1,000,000. The impression is at once +conveyed of a considerable degree of wealth, but it is +rather from the fact that that fortune represents an +annual income of $40,000 than, from the actual magnitude +<a class="pgnum" id="page-35" title="- 35 -"></a>of the fortune itself. The number 1,000,000 is, in itself, +so greatly in excess of anything that enters into our daily +experience that we have but a vague conception of it, +except as something very great. We are not, after all, +so very much better off than the child who, with his arms +about his mother's neck, informs her with perfect gravity +and sincerity that he “loves her a million bushels.” His +idea is merely of some very great amount, and our own +is often but little clearer when we use the expressions +which are so easily represented by a few digits. Among +the uneducated portions of civilized communities the +limit of clear comprehension of number is not only relatively, +but absolutely, very low. Travellers in Russia +have informed the writer that the peasants of that +country have no distinct idea of a number consisting of +but a few hundred even. There is no reason to doubt +this testimony. The entire life of a peasant might be +passed without his ever having occasion to use a number +as great as 500, and as a result he might have respecting +that number an idea less distinct than a trained mathematician +would have of the distance from the earth to +the sun. De Quincey<a id="FNA-50" href="#FN-50"><sup>50</sup></a> incidentally mentions this characteristic +in narrating a conversation which occurred +while he was at Carnarvon, a little town in Wales. “It +was on this occasion,” he says, “that I learned how vague +are the ideas of number in unpractised minds. ‘What +number of people do you think,’ I said to an elderly +person, ‘will be assembled this day at Carnarvon?’ +<a class="pgnum" id="page-36" title="- 36 -"></a>‘What number?’ rejoined the person addressed; ‘what +number? Well, really, now, I should reckon—perhaps +a matter of four million.’ Four millions of <i>extra</i> people +in little Carnarvon, that could barely find accommodation +(I should calculate) for an extra four hundred!” +So the Eskimo and the South American Indian are, +after all, not so very far behind the “elderly person” +of Carnarvon, in the distinct perception of a number +which familiarity renders to us absurdly small.</p> + + + + +<h2><a class="pgnum" id="page-37" title="- 37 -"></a>Chapter III.</h2> +<h3>The Origin of Number Words.</h3> + + +<p><span class="initial">In</span> the comparison of languages and the search for +primitive root forms, no class of expressions has been +subjected to closer scrutiny than the little cluster of +words, found in each language, which constitutes a part +of the daily vocabulary of almost every human being—the +words with which we begin our counting. It is +assumed, and with good reason, that these are among +the earlier words to appear in any language; and in the +mutations of human speech, they are found to suffer less +than almost any other portion of a language. Kinship +between tongues remote from each other has in many +instances been detected by the similarity found to exist +among the every-day words of each; and among these +words one may look with a good degree of certainty +for the 1, 2, 3, etc., of the number scale. So fruitful +has been this line of research, that the attempt has been +made, even, to establish a common origin for all the +races of mankind by means of a comparison of numeral +words.<a id="FNA-51" href="#FN-51"><sup>51</sup></a> But in this instance, as in so many others that +will readily occur to the mind, the result has been that +<a class="pgnum" id="page-38" title="- 38 -"></a>the theory has finally taken possession of the author and +reduced him to complete subjugation, instead of remaining +his servant and submitting to the legitimate results +of patient and careful investigation. Linguistic research +is so full of snares and pitfalls that the student must +needs employ the greatest degree of discrimination +before asserting kinship of race because of resemblances +in vocabulary; or even relationship between words in +the same language because of some chance likeness of +form that may exist between them. Probably no one +would argue that the English and the Babusessé of +Central Africa were of the same primitive stock simply +because in the language of the latter <i>five atano</i> +means 5, and <i>ten kumi</i> means 10.<a id="FNA-52" href="#FN-52"><sup>52</sup></a> But, on the other +hand, many will argue that, because the German <i>zehn</i> +means 10, and <i>zehen</i> means toes, the ancestors of +the Germans counted on their toes; and that with +them, 10 was the complete count of the toes. It +may be so. We certainly have no evidence with +which to disprove this; but, before accepting it as a +fact, or even as a reasonable hypothesis, we may be +pardoned for demanding some evidence aside from the +mere resemblance in the form of the words. If, in +the study of numeral words, form is to constitute our +chief guide, we must expect now and then to be +confronted with facts which are not easily reconciled +with any pet theory.</p> + +<p>The scope of the present work will admit of no +<a class="pgnum" id="page-39" title="- 39 -"></a>more than a hasty examination of numeral forms, in +which only actual and well ascertained meanings will +be considered. But here we are at the outset confronted +with a class of words whose original meanings +appear to be entirely lost. They are what may be +termed the numerals proper—the native, uncompounded +words used to signify number. Such words +are the one, two, three, etc., of English; the eins, +zwei, drei, etc., of German; words which must at +some time, in some prehistoric language, have had +definite meanings entirely apart from those which they +now convey to our minds. In savage languages it is +sometimes possible to detect these meanings, and thus +to obtain possession of the clue that leads to the +development, in the barbarian's rude mind, of a count +scale—a number system. But in languages like those +of modern Europe, the pedigree claimed by numerals +is so long that, in the successive changes through +which they have passed, all trace of their origin seems +to have been lost.</p> + +<p>The actual number of such words is, however, surprisingly +small in any language. In English we count +by simple words only to 10. From this point onward +all our numerals except “hundred” and “thousand” +are compounds and combinations of the names of +smaller numbers. The words we employ to designate +the higher orders of units, as million, billion, trillion, +etc., are appropriated bodily from the Italian; and the +native words <i>pair</i>, <i>tale</i>, <i>brace</i>, <i>dozen</i>, <i>gross</i>, and <i>score</i>, +<a class="pgnum" id="page-40" title="- 40 -"></a>can hardly be classed as numerals in the strict sense of +the word. German possesses exactly the same number +of native words in its numeral scale as English; and the +same may be said of the Teutonic languages generally, +as well as of the Celtic, the Latin, the Slavonic, and +the Basque. This is, in fact, the universal method +observed in the formation of any numeral scale, though +the actual number of simple words may vary. The +Chiquito language has but one numeral of any kind +whatever; English contains twelve simple terms; Sanskrit +has twenty-seven, while Japanese possesses twenty-four, +and the Chinese a number almost equally great. +Very many languages, as might be expected, contain +special numeral expressions, such as the German <i>dutzend</i> +and the French <i>dizaine</i>; but these, like the English +<i>dozen</i> and <i>score</i>, are not to be regarded as numerals +proper.</p> + +<p>The formation of numeral words shows at a glance +the general method in which any number scale has +been built up. The primitive savage counts on his +fingers until he has reached the end of one, or more +probably of both, hands. Then, if he wishes to proceed +farther, some mark is made, a pebble is laid aside, a +knot tied, or some similar device employed to signify +that all the counters at his disposal have been used. +Then the count begins anew, and to avoid multiplication +of words, as well as to assist the memory, the +terms already used are again resorted to; and the name +by which the first halting-place was designated is repeated +<a class="pgnum" id="page-41" title="- 41 -"></a>with each new numeral. Hence the thirteen, +fourteen, fifteen, etc., which are contractions of the +fuller expressions three-and-ten, four-and-ten, five-and-ten, +etc. The specific method of combination may not +always be the same, as witness the <i>eighteen</i>, or eight-ten, +in English, and <i>dix-huit,</i> or ten-eight, in French; +<i>forty-five</i>, or four-tens-five, in English, and <i>fünf und +vierzig</i>, or five and four tens in German. But the +general method is the same the world over, presenting +us with nothing but local variations, which are, relatively +speaking, entirely unimportant. With this fact +in mind, we can cease to wonder at the small number +of simple numerals in any language. It might, indeed, +be queried, why do any languages, English and German, +for example, have unusual compounds for 11 and 12? +It would seem as though the regular method of compounding +should begin with 10 and 1, instead of 10 +and 3, in any language using a system with 10 as +a base. An examination of several hundred numeral +scales shows that the Teutonic languages are somewhat +exceptional in this respect. The words <i>eleven</i> and +<i>twelve</i> are undoubtedly combinations, but not in the +same direct sense as <i>thirteen</i>, <i>twenty-five</i>, etc. The +same may be said of the French <i>onze</i>, <i>douze</i>, <i>treize</i>, +<i>quatorze</i>, <i>quinze</i>, and <i>seize</i>, which are obvious compounds, +but not formed in the same manner as the +numerals above that point. Almost all civilized languages, +however, except the Teutonic, and practically +all uncivilized languages, begin their direct numeral +<a class="pgnum" id="page-42" title="- 42 -"></a>combinations as soon as they have passed their number +base, whatever that may be. To give an illustration, +selected quite at random from among the barbarous +tribes of Africa, the Ki-Swahili numeral scale runs as +follows:<a id="FNA-53" href="#FN-53"><sup>53</sup></a></p> + +<table class="scale"> +<tr><td>1.</td><td>moyyi,</td></tr> +<tr><td>2.</td><td>mbiri,</td></tr> +<tr><td>3.</td><td>tato,</td></tr> +<tr><td>4.</td><td>ena,</td></tr> +<tr><td>5.</td><td>tano,</td></tr> +<tr><td>6.</td><td>seta,</td></tr> +<tr><td>7.</td><td>saba,</td></tr> +<tr><td>8.</td><td>nani,</td></tr> +<tr><td>9.</td><td>kenda,</td></tr> +<tr><td>10.</td><td>kumi,</td></tr> +<tr><td>11.</td><td>kumi na moyyi,</td></tr> +<tr><td>12.</td><td>kumi na mbiri,</td></tr> +<tr><td>13.</td><td>kumi na tato,</td></tr> +<tr><td></td><td class="alt">etc.</td></tr> + +</table> + +<p>The words for 11, 12, and 13, are seen at a glance to +signify ten-and-one, ten-and-two, ten-and-three, and the +count proceeds, as might be inferred, in a similar +manner as far as the number system extends. Our +English combinations are a little closer than these, and +the combinations found in certain other languages are, in +turn, closer than those of the English; as witness the +<i>once</i>, 11, <i>doce</i>, 12, <i>trece</i>, 13, etc., of Spanish. But the +process is essentially the same, and the law may be +accepted as practically invariable, that all numerals +greater than the base of a system are expressed by +compound words, except such as are necessary to establish +some new order of unit, as hundred or thousand.</p> + +<p>In the scale just given, it will be noticed that the +larger number precedes the smaller, giving 10 + 1, 10 + 2, +etc., instead of 1 + 10, 2 + 10, etc. This seems entirely +<a class="pgnum" id="page-43" title="- 43 -"></a>natural, and hardly calls for any comment whatever. +But we have only to consider the formation of our +English “teens” to see that our own method is, at +its inception, just the reverse of this. Thirteen, 14, +and the remaining numerals up to 19 are formed by +prefixing the smaller number to the base; and it is +only when we pass 20 that we return to the more direct +and obvious method of giving precedence to the larger. +In German and other Teutonic languages the inverse +method is continued still further. Here 25 is <i>fünf und +zwanzig</i>, 5 and 20; 92 is <i>zwei und neunzig</i>, 2 and 90, +and so on to 99. Above 100 the order is made direct, +as in English. Of course, this mode of formation +between 20 and 100 is permissible in English, where +“five and twenty” is just as correct a form as twenty-five. +But it is archaic, and would soon pass out of the +language altogether, were it not for the influence of +some of the older writings which have had a strong +influence in preserving for us many of older and more +essentially Saxon forms of expression.</p> + +<p>Both the methods described above are found in all +parts of the world, but what I have called the direct +is far more common than the other. In general, where +the smaller number precedes the larger it signifies +multiplication instead of addition. Thus, when we say +“thirty,” <i>i.e.</i> three-ten, we mean 3 × 10; just as “three +hundred” means 3 × 100. When the larger precedes +the smaller, we must usually understand addition. But +to both these rules there are very many exceptions. +<a class="pgnum" id="page-44" title="- 44 -"></a>Among higher numbers the inverse order is very rarely +used; though even here an occasional exception is found. +The Taensa Indians, for example, place the smaller +numbers before the larger, no matter how far their +scale may extend. To say 1881 they make a complete +inversion of our own order, beginning with 1 and ending +with 1000. Their full numeral for this is <i>yeha av +wabki mar-u-wab mar-u-haki</i>, which means, literally, +1 + 80 + 100 × 8 + 100 × 10.<a id="FNA-54" href="#FN-54"><sup>54</sup></a> Such exceptions are, however, +quite rare.</p> + +<p>One other method of combination, that of subtraction, +remains to be considered. Every student of Latin +will recall at once the <i>duodeviginti</i>, 2 from 20, and +<i>undeviginti</i>, 1 from 20, which in that language are the +regular forms of expression for 18 and 19. At first +they seem decidedly odd; but familiarity soon accustoms +one to them, and they cease entirely to attract +any special attention. This principle of subtraction, +which, in the formation of numeral words, is quite +foreign to the genius of English, is still of such common +occurrence in other languages that the Latin +examples just given cease to be solitary instances.</p> + +<p>The origin of numerals of this class is to be found +in the idea of reference, not necessarily to the last, but +to the nearest, halting-point in the scale. Many tribes +seem to regard 9 as “almost 10,” and to give it a name +which conveys this thought. In the Mississaga, one of +<a class="pgnum" id="page-45" title="- 45 -"></a>the numerous Algonquin languages, we have, for example, +the word <i>cangaswi</i>, “incomplete 10,” for 9.<a id="FNA-55" href="#FN-55"><sup>55</sup></a> In the +Kwakiutl of British Columbia, 8 as well as 9 is formed +in this way; these two numbers being <i>matlguanatl</i>, +10 − 2, and <i>nanema</i>, 10 − 1, respectively.<a id="FNA-56" href="#FN-56"><sup>56</sup></a> In many of +the languages of British Columbia we find a similar +formation for 8 and 9, or for 9 alone. The same formation +occurs in Malay, resulting in the numerals <i>delapan</i>, +10 − 2, and <i>sambilan</i> 10 − 1.<a id="FNA-57" href="#FN-57"><sup>57</sup></a> In Green Island, one of +the New Ireland group, these become simply <i>andra-lua</i>, +“less 2,” and <i>andra-si</i>, “less 1.”<a id="FNA-58" href="#FN-58"><sup>58</sup></a> In the Admiralty +Islands this formation is carried back one step further, +and not only gives us <i>shua-luea</i>, “less 2,” and <i>shu-ri</i>, “less +1,” but also makes 7 appear as <i>sua-tolu</i>, “less 3.”<a id="FNA-59" href="#FN-59"><sup>59</sup></a> Surprising +as this numeral is, it is more than matched by +the Ainu scale, which carries subtraction back still +another step, and calls 6, 10 − 4. The four numerals from +6 to 9 in this scale are respectively, <i>iwa</i>, 10 − 4, <i>arawa</i>, +10 − 3, <i>tupe-san</i>, 10 − 2, and <i>sinepe-san</i>, 10 − 1.<a id="FNA-60" href="#FN-60"><sup>60</sup></a> Numerous +examples of this kind of formation will be found in +later chapters of this work; but they will usually be +found to occur in one or both of the numerals, 8 and 9. +Occasionally they appear among the higher numbers; +as in the Maya languages, where, for example, 99 years +<a class="pgnum" id="page-46" title="- 46 -"></a>is “one single year lacking from five score years,”<a id="FNA-61" href="#FN-61"><sup>61</sup></a> +and in the Arikara dialects, where 98 and 99 are “5 +men minus” and “5 men 1 not.”<a id="FNA-62" href="#FN-62"><sup>62</sup></a> The Welsh, Danish, +and other languages less easily accessible than these to +the general student, also furnish interesting examples +of a similar character.</p> + +<p>More rarely yet are instances met with of languages +which make use of subtraction almost as freely as addition, +in the composition of numerals. Within the +past few years such an instance has been noticed in +the case of the Bellacoola language of British Columbia. +In their numeral scale 15, “one foot,” is followed +by 16, “one man less 4”; 17, “one man less 3”; 18, +“one man less 2”; 19, “one man less 1”; and 20, one +man. Twenty-five is “one man and one hand”; 26, “one +man and two hands less 4”; 36, “two men less 4”; and +so on. This method of formation prevails throughout +the entire numeral scale.<a id="FNA-63" href="#FN-63"><sup>63</sup></a></p> + +<p>One of the best known and most interesting examples +of subtraction as a well-defined principle of formation +is found in the Maya scale. Up to 40 no special +peculiarity appears; but as the count progresses beyond +that point we find a succession of numerals which one +is almost tempted to call 60 − 19, 60 − 18, 60 − 17, etc. +Literally translated the meanings seem to be 1 to 60, +2 to 60, 3 to 60, etc. The point of reference is 60, +<a class="pgnum" id="page-47" title="- 47 -"></a>and the thought underlying the words may probably +be expressed by the paraphrases, “1 on the third score, +2 on the third score, 3 on the third score,” etc. Similarly, +61 is 1 on the fourth score, 81 is one on the +fifth score, 381 is 1 on the nineteenth score, and so on +to 400. At 441 the same formation reappears; and it +continues to characterize the system in a regular and +consistent manner, no matter how far it is extended.<a id="FNA-64" href="#FN-64"><sup>64</sup></a></p> + +<p>The Yoruba language of Africa is another example +of most lavish use of subtraction; but it here results +in a system much less consistent and natural than that +just considered. Here we find not only 5, 10, and 20 +subtracted from the next higher unit, but also 40, and +even 100. For example, 360 is 400 − 40; 460 is 500 − 40; +500 is 600 − 100; 1300 is 1400 − 100, etc. One of the +Yoruba units is 200; and all the odd hundreds up to +2000, the next higher unit, are formed by subtracting +100 from the next higher multiple of 200. The system +is quite complex, and very artificial; and seems to +have been developed by intercourse with traders.<a id="FNA-65" href="#FN-65"><sup>65</sup></a></p> + +<p>It has already been stated that the primitive meanings +of our own simple numerals have been lost. This +is also true of the languages of nearly all other civilized +peoples, and of numerous savage races as well. +We are at liberty to suppose, and we do suppose, that +in very many cases these words once expressed meanings +<a class="pgnum" id="page-48" title="- 48 -"></a>closely connected with the names of the fingers, or +with the fingers themselves, or both. Now and then a +case is met with in which the numeral word frankly +avows its meaning—as in the Botocudo language, +where 1 is expressed by <i>podzik</i>, finger, and 2 by <i>kripo</i>, +double finger;<a id="FNA-66" href="#FN-66"><sup>66</sup></a> and in the Eskimo dialect of Hudson's +Bay, where <i>eerkitkoka</i> means both 10 and little finger.<a id="FNA-67" href="#FN-67"><sup>67</sup></a> +Such cases are, however, somewhat exceptional.</p> + +<p>In a few noteworthy instances, the words composing +the numeral scale of a language have been carefully +investigated and their original meanings accurately +determined. The simple structure of many of the rude +languages of the world should render this possible in a +multitude of cases; but investigators are too often content +with the mere numerals themselves, and make no +inquiry respecting their meanings. But the following +exposition of the Zuñi scale, given by Lieutenant +Gushing<a id="FNA-68" href="#FN-68"><sup>68</sup></a> leaves nothing to be desired:</p> + +<table class="scale"> +<tr><td>1.</td><td>töpinte</td><td>= taken to start with.</td></tr> +<tr><td>2.</td><td>kwilli</td><td>= put down together with.</td></tr> +<tr><td>3.</td><td>ha'ī</td><td>= the equally dividing finger.</td></tr> +<tr><td>4.</td><td>awite</td><td>= all the fingers all but done with.</td></tr> +<tr><td>5.</td><td>öpte</td><td>= the notched off.</td></tr> +</table> + +<p>This finishes the list of original simple numerals, +the Zuñi stopping, or “notching off,” when he finishes +the fingers of one hand. Compounding now begins.</p> + +<table class="scale"> +<tr><td><a class="pgnum" id="page-49" title="- 49 -"></a>6.</td><td>topalïk'ya</td><td>= another brought to add to the done with.</td></tr> +<tr><td>7.</td><td>kwillilïk'ya</td><td>= two brought to and held up with the rest.</td></tr> +<tr><td>8.</td><td>hailïk'ye</td><td>= three brought to and held up with the rest.</td></tr> +<tr><td>9.</td><td>tenalïk'ya</td><td>= all but all are held up with the rest.</td></tr> +<tr><td>10.</td><td>ästem'thila</td><td>= all the fingers.</td></tr> +<tr><td>11.</td><td>ästem'thla topayä'thl'tona</td><td>= all the fingers and another over above held.</td></tr> +</table> + +<p class="noindent">The process of formation indicated in 11 is used in the +succeeding numerals up to 19.</p> + +<table class="scale"> +<tr><td>20.</td><td>kwillik'yënästem'thlan</td><td>= two times all the fingers.</td></tr> +<tr><td>100.</td><td>ässiästem'thlak'ya</td><td>= the fingers all the fingers.</td></tr> +<tr><td>1000.</td><td>ässiästem'thlanak'yënästem'thla</td><td>= the fingers all the fingers times all the fingers.</td></tr> +</table> + +<p class="noindent">The only numerals calling for any special note are +those for 11 and 9. For 9 we should naturally expect +a word corresponding in structure and meaning to the +words for 7 and 8. But instead of the “four brought +to and held up with the rest,” for which we naturally +look, the Zuñi, to show that he has used all of his +fingers but one, says “all but all are held up with the +rest.” To express 11 he cannot use a similar form of +composition, since he has already used it in constructing +his word for 6, so he says “all the fingers and +another over above held.”</p> + +<p>The one remarkable point to be noted about the +Zuñi scale is, after all, the formation of the words for +1 and 2. While the savage almost always counts on +his fingers, it does not seem at all certain that these +words would necessarily be of finger formation. The +<a class="pgnum" id="page-50" title="- 50 -"></a>savage can always distinguish between one object and +two objects, and it is hardly reasonable to believe that +any external aid is needed to arrive at a distinct perception +of this difference. The numerals for 1 and 2 +would be the earliest to be formed in any language, +and in most, if not all, cases they would be formed +long before the need would be felt for terms to +describe any higher number. If this theory be correct, +we should expect to find finger names for numerals +beginning not lower than 3, and oftener with 5 +than with any other number. The highest authority +has ventured the assertion that all numeral words have +their origin in the names of the fingers;<a id="FNA-69" href="#FN-69"><sup>69</sup></a> substantially +the same conclusion was reached by Professor Pott, of +Halle, whose work on numeral nomenclature led him +deeply into the study of the origin of these words. +But we have abundant evidence at hand to show that, +universal as finger counting has been, finger origin for +numeral words has by no means been universal. That +it is more frequently met with than any other origin is +unquestionably true; but in many instances, which will +be more fully considered in the following chapter, we +find strictly non-digital derivations, especially in the +case of the lowest members of the scale. But in nearly +all languages the origin of the words for 1, 2, 3, and +4 are so entirely unknown that speculation respecting +them is almost useless.</p> + +<p>An excellent illustration of the ordinary method of +<a class="pgnum" id="page-51" title="- 51 -"></a>formation which obtains among number scales is furnished +by the Eskimos of Point Barrow,<a id="FNA-70" href="#FN-70"><sup>70</sup></a> who have pure +numeral words up to 5, and then begin a systematic +course of word formation from the names of their +fingers. If the names of the first five numerals are of +finger origin, they have so completely lost their original +form, or else the names of the fingers themselves have +so changed, that no resemblance is now to be detected +between them. This scale is so interesting that it is +given with considerable fulness, as follows:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">atauzik.</td></tr> +<tr><td>2.</td><td colspan="2">madro.</td></tr> +<tr><td>3.</td><td colspan="2">pinasun.</td></tr> +<tr><td>4.</td><td colspan="2">sisaman.</td></tr> +<tr><td>5.</td><td colspan="2">tudlemut.</td></tr> +<tr><td>6.</td><td>atautyimin akbinigin [tudlimu(t)]</td><td>= 5 and 1 on the next.</td></tr> +<tr><td>7.</td><td>madronin akbinigin</td><td>= twice on the next.</td></tr> +<tr><td>8.</td><td>pinasunin akbinigin</td><td>= three times on the next.</td></tr> +<tr><td>9.</td><td>kodlinotaila</td><td>= that which has not its 10.</td></tr> +<tr><td>10.</td><td>kodlin</td><td>= the upper part—<i>i.e.</i> the fingers.</td></tr> +<tr><td>14.</td><td>akimiaxotaityuna</td><td>= I have not 15.</td></tr> +<tr><td>15.</td><td colspan="2">akimia. [This seems to be a real numeral word.]</td></tr> +<tr><td>20.</td><td>inyuina</td><td>= a man come to an end.</td></tr> +<tr><td>25.</td><td>inyuina tudlimunin akbinidigin</td><td>= a man come to an end and 5 on the next.</td></tr> +<tr><td>30.</td><td>inyuina kodlinin akbinidigin</td><td>= a man come to an end and 10 on the next.</td></tr> +<tr><td>35.</td><td>inyuina akimiamin aipalin</td><td>= a man come to an end accompanied by 1 fifteen times.</td></tr> +<tr><td>40.</td><td>madro inyuina</td><td>= 2 men come to an end.</td></tr> +</table> + +<p class="noindent">In this scale we find the finger origin appearing so +clearly and so repeatedly that one feels some degree of +surprise at finding 5 expressed by a pure numeral instead +of by some word meaning <i>hand</i> or <i>fingers of one +<a class="pgnum" id="page-52" title="- 52 -"></a>hand</i>. In this respect the Eskimo dialects are somewhat +exceptional among scales built up of digital words. +The system of the Greenland Eskimos, though differing +slightly from that of their Point Barrow cousins, shows +the same peculiarity. The first ten numerals of this +scale are:<a id="FNA-71" href="#FN-71"><sup>71</sup></a></p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">atausek.</td></tr> +<tr><td>2.</td><td colspan="2">mardluk.</td></tr> +<tr><td>3.</td><td colspan="2">pingasut.</td></tr> +<tr><td>4.</td><td colspan="2">sisamat.</td></tr> +<tr><td>5.</td><td colspan="2">tatdlimat.</td></tr> +<tr><td>6.</td><td>arfinek-atausek</td><td>= to the other hand 1.</td></tr> +<tr><td>7.</td><td>arfinek-mardluk</td><td>= to the other hand 2.</td></tr> +<tr><td>8.</td><td>arfinek-pingasut</td><td>= to the other hand 3.</td></tr> +<tr><td>9.</td><td>arfinek-sisamat</td><td>= to the other hand 4.</td></tr> +<tr><td>10.</td><td colspan="2">kulit.</td></tr> +</table> + +<p class="noindent">The same process is now repeated, only the feet instead +of the hands are used; and the completion of the +second 10 is marked by the word <i>innuk</i>, man. It may +be that the Eskimo word for 5 is, originally, a digital +word, but if so, the fact has not yet been detected. +From the analogy furnished by other languages we are +justified in suspecting that this may be the case; for +whenever a number system contains digital words, we +expect them to begin with <i>five</i>, as, for example, in the +Arawak scale,<a id="FNA-72" href="#FN-72"><sup>72</sup></a> which runs:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">abba.</td></tr> +<tr><td>2.</td><td colspan="2">biama.</td></tr> +<tr><td>3.</td><td colspan="2">kabbuhin.</td></tr> +<tr><td>4.</td><td colspan="2">bibiti.</td></tr> +<tr><td>5.</td><td>abbatekkábe</td><td>= 1 hand.</td></tr> +<tr><td>6.</td><td>abbatiman</td><td>= 1 of the other.</td></tr> +<tr><td>7.</td><td>biamattiman</td><td>= 2 of the other.</td></tr> +<tr><td>8.</td><td>kabbuhintiman</td><td>= 3 of the other.</td></tr> +<tr><td><a class="pgnum" id="page-53" title="- 53 -"></a>9.</td><td>bibitiman</td><td>= 4 of the other.</td></tr> +<tr><td>10.</td><td>biamantekábbe</td><td>= 2 hands.</td></tr> +<tr><td>11.</td><td>abba kutihibena</td><td>= 1 from the feet.</td></tr> +<tr><td>20.</td><td>abba lukku</td><td>= hands feet.</td></tr> +</table> + +<p>The four sets of numerals just given may be regarded +as typifying one of the most common forms of +primitive counting; and the words they contain serve +as illustrations of the means which go to make up the +number scales of savage races. Frequently the finger +and toe origin of numerals is perfectly apparent, as +in the Arawak system just given, which exhibits the +simplest and clearest possible method of formation. +Another even more interesting system is that of the +Montagnais of northern Canada.<a id="FNA-73" href="#FN-73"><sup>73</sup></a> Here, as in the Zuñi +scale, the words are digital from the outset.</p> + +<table class="scale"> +<tr><td>1.</td><td></td><td>inl'are</td><td>= the end is bent.</td></tr> +<tr><td>2.</td><td></td><td>nak'e</td><td>= another is bent.</td></tr> +<tr><td>3.</td><td></td><td>t'are</td><td>= the middle is bent.</td></tr> +<tr><td>4.</td><td></td><td>dinri</td><td>= there are no more except this.</td></tr> +<tr><td>5.</td><td></td><td>se-sunla-re</td><td>= the row on the hand.</td></tr> +<tr><td>6.</td><td></td><td>elkke-t'are</td><td>= 3 from each side.</td></tr> +<tr><td rowspan="2" style="vertical-align:middle;padding:0">7.</td><td rowspan="2" style="vertical-align:middle;padding:0"><span class="twoline">{</span></td><td>t'a-ye-oyertan</td><td>= there are still 3 of them.</td></tr> +<tr><td style="text-align:left">inl'as dinri</td><td>= on one side there are 4 of them.</td></tr> +<tr><td>8.</td><td></td><td>elkke-dinri</td><td>= 4 on each side.</td></tr> +<tr><td>9.</td><td></td><td>inl'a-ye-oyert'an</td><td>= there is still 1 more.</td></tr> +<tr><td>10.</td><td></td><td>onernan</td><td>= finished on each side.</td></tr> +<tr><td>11.</td><td></td><td>onernan inl'are ttcharidhel</td><td>= 1 complete and 1.</td></tr> +<tr><td>12.</td><td></td><td>onernan nak'e ttcharidhel</td><td>= 1 complete and 2, etc.</td></tr> +</table> + +<p class="noindent">The formation of 6, 7, and 8 of this scale is somewhat +different from that ordinarily found. To express 6, the +<a class="pgnum" id="page-54" title="- 54 -"></a>Montagnais separates the thumb and forefinger from +the three remaining fingers of the left hand, and bringing +the thumb of the right hand close to them, says: +“3 from each side.” For 7 he either subtracts from +10, saying: “there are still 3 of them,” or he brings +the thumb and forefinger of the right hand up to the +thumb of the left, and says: “on one side there are 4 +of them.” He calls 8 by the same name as many of +the other Canadian tribes, that is, two 4's; and to show +the proper number of fingers, he closes the thumb and +little finger of the right hand, and then puts the three +remaining fingers beside the thumb of the left hand. +This method is, in some of these particulars, different +from any other I have ever examined.</p> + +<p>It often happens that the composition of numeral +words is less easily understood, and the original meanings +more difficult to recover, than in the examples +already given. But in searching for number systems +which show in the formation of their words the influence +of finger counting, it is not unusual to find those +in which the derivation from native words signifying +<i>finger, hand, toe, foot</i>, and <i>man</i>, is just as frankly obvious +as in the case of the Zuñi, the Arawak, the Eskimo, +or the Montagnais scale. Among the Tamanacs,<a id="FNA-74" href="#FN-74"><sup>74</sup></a> one +of the numerous Indian tribes of the Orinoco, the numerals +are as strictly digital as in any of the systems +already examined. The general structure of the Tamanac +scale is shown by the following numerals:</p> + + +<table class="scale"> +<tr><td><a class="pgnum" id="page-55" title="- 55 -"></a>5.</td><td>amgnaitone</td><td>= 1 hand complete.</td></tr> +<tr><td>6.</td><td>itacono amgna pona tevinitpe</td><td>= 1 on the other hand.</td></tr> +<tr><td>10.</td><td>amgna aceponare</td><td>= all of the 2 hands.</td></tr> +<tr><td>11.</td><td>puitta pona tevinitpe</td><td>= 1 on the foot.</td></tr> +<tr><td>16.</td><td>itacono puitta pona tevinitpe</td><td>= 1 on the other foot.</td></tr> +<tr><td>20.</td><td>tevin itoto</td><td>= 1 man.</td></tr> +<tr><td>21.</td><td>itacono itoto jamgnar bona tevinitpe</td><td>= 1 on the hands of another man.</td></tr> +</table> + +<p>In the Guarani<a id="FNA-75" href="#FN-75"><sup>75</sup></a> language of Paraguay the same +method is found, with a different form of expression +for 20. Here the numerals in question are</p> + +<table class="scale"> +<tr><td>5.</td><td>asepopetei</td><td>= one hand.</td></tr> +<tr><td>10.</td><td>asepomokoi</td><td>= two hands.</td></tr> +<tr><td>20.</td><td>asepo asepi abe</td><td>= hands and feet.</td></tr> +</table> + +<p>Another slight variation is furnished by the Kiriri +language,<a id="FNA-76" href="#FN-76"><sup>76</sup></a> which is also one of the numerous South +American Indian forms of speech, where we find the +words to be</p> + +<table class="scale"> +<tr><td>5.</td><td>mi biche misa</td><td>= one hand.</td></tr> +<tr><td>10.</td><td>mikriba misa sai</td><td>= both hands.</td></tr> +<tr><td>20.</td><td>mikriba misa idecho ibi sai</td><td>= both hands together with the feet.</td></tr> +</table> + +<p>Illustrations of this kind might be multiplied almost +indefinitely; and it is well to note that they may be +drawn from all parts of the world. South America is +peculiarly rich in native numeral words of this kind; +and, as the examples above cited show, it is the field +to which one instinctively turns when this subject is +under discussion. The Zamuco numerals are, among +<a class="pgnum" id="page-56" title="- 56 -"></a>others, exceedingly interesting, giving us still a new +variation in method. They are<a id="FNA-77" href="#FN-77"><sup>77</sup></a></p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">tsomara.</td></tr> +<tr><td>2.</td><td colspan="2">gar.</td></tr> +<tr><td>3.</td><td colspan="2">gadiok.</td></tr> +<tr><td>4.</td><td colspan="2">gahagani.</td></tr> +<tr><td>5.</td><td>tsuena yimana-ite</td><td>= ended 1 hand.</td></tr> +<tr><td>6.</td><td>tsomara-hi</td><td>= 1 on the other.</td></tr> +<tr><td>7.</td><td>gari-hi</td><td>= 2 on the other.</td></tr> +<tr><td>8.</td><td>gadiog-ihi</td><td>= 3 on the other.</td></tr> +<tr><td>9.</td><td>gahagani-hi</td><td>= 4 on the other.</td></tr> +<tr><td>10.</td><td>tsuena yimana-die</td><td>= ended both hands.</td></tr> +<tr><td>11.</td><td>tsomara yiri-tie</td><td>= 1 on the foot.</td></tr> +<tr><td>12.</td><td>gar yiritie</td><td>= 2 on the foot.</td></tr> +<tr><td>20.</td><td>tsuena yiri-die</td><td>= ended both feet.</td></tr> +</table> + +<p>As is here indicated, the form of progression from +5 to 10, which we should expect to be “hand-1,” or +“hand-and-1,” or some kindred expression, signifying +that one hand had been completed, is simply “1 on the +other.” Again, the expressions for 11, 12, etc., are +merely “1 on the foot,” “2 on the foot,” etc., while 20 +is “both feet ended.”</p> + +<p>An equally interesting scale is furnished by the language +of the Maipures<a id="FNA-78" href="#FN-78"><sup>78</sup></a> of the Orinoco, who count</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">papita.</td></tr> +<tr><td>2.</td><td colspan="2">avanume.</td></tr> +<tr><td>3.</td><td colspan="2">apekiva.</td></tr> +<tr><td>4.</td><td colspan="2">apekipaki.</td></tr> +<tr><td>5.</td><td>papitaerri capiti</td><td>= 1 only hand.</td></tr> +<tr><td>6.</td><td>papita yana pauria capiti purena</td><td>= 1 of the other hand we take.</td></tr> +<tr><td>10.</td><td>apanumerri capiti</td><td>= 2 hands.</td></tr> +<tr><td>11.</td><td>papita yana kiti purena</td><td>= 1 of the toes we take.</td></tr> +<tr><td>20.</td><td>papita camonee</td><td>= 1 man.</td></tr> +<tr><td>40.</td><td>avanume camonee</td><td>= 2 men.</td></tr> +<tr><td>60.</td><td>apekiva camonee</td><td>= 3 men, etc.</td></tr> +</table> + +<p><a class="pgnum" id="page-57" title="- 57 -"></a>In all the examples thus far given, 20 is expressed +either by the equivalent of “man” or by some formula +introducing the word “feet.” Both these modes of expressing +what our own ancestors termed a “score,” are +so common that one hesitates to say which is of the +more frequent use. The following scale, from one of +the Betoya dialects<a id="FNA-79" href="#FN-79"><sup>79</sup></a> of South America, is quite remarkable +among digital scales, making no use of either +“man” or “foot,” but reckoning solely by fives, or +hands, as the numerals indicate.</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">tey.</td></tr> +<tr><td>2.</td><td colspan="2">cayapa.</td></tr> +<tr><td>3.</td><td colspan="2">toazumba.</td></tr> +<tr><td>4.</td><td>cajezea</td><td>= 2 with plural termination.</td></tr> +<tr><td>5.</td><td>teente</td><td>= hand.</td></tr> +<tr><td>6.</td><td>teyentetey</td><td>= hand + 1.</td></tr> +<tr><td>7.</td><td>teyente cayapa</td><td>= hand + 2.</td></tr> +<tr><td>8.</td><td>teyente toazumba</td><td>= hand + 3.</td></tr> +<tr><td>9.</td><td>teyente caesea</td><td>= hand + 4.</td></tr> +<tr><td>10.</td><td>caya ente, or caya huena</td><td>= 2 hands.</td></tr> +<tr><td>11.</td><td>caya ente-tey</td><td>= 2 hands + 1.</td></tr> +<tr><td>15.</td><td>toazumba-ente</td><td>= 3 hands.</td></tr> +<tr><td>16.</td><td>toazumba-ente-tey</td><td>= 3 hands + 1.</td></tr> +<tr><td>20.</td><td>caesea ente</td><td>= 4 hands.</td></tr> +</table> + + +<p>In the last chapter mention was made of the scanty +numeral systems of the Australian tribes, but a single +scale was alluded to as reaching the comparatively high +limit of 20. This system is that belonging to the Pikumbuls,<a id="FNA-80" href="#FN-80"><sup>80</sup></a> +and the count runs thus:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">mal.</td></tr> +<tr><td>2.</td><td colspan="2">bular.</td></tr> +<tr><td>3.</td><td colspan="2">guliba.</td></tr> +<tr><td>4.</td><td>bularbular</td><td>= 2-2.</td></tr> +<tr><td>5.</td><td colspan="2">mulanbu.</td></tr> +<tr><td>6.</td><td>malmulanbu mummi</td><td>= 1 and 5 added on.</td></tr> +<tr><td><a class="pgnum" id="page-58" title="- 58 -"></a>7.</td><td>bularmulanbu mummi</td><td>= 2 and 5 added on.</td></tr> +<tr><td>8.</td><td>gulibamulanbu mummi</td><td>= 3 and 5 added on.</td></tr> +<tr><td>9.</td><td>bularbularmulanbu mummi</td><td>= 4 and 5 added on.</td></tr> +<tr><td>10.</td><td>bularin murra</td><td>= belonging to the 2 hands.</td></tr> +<tr><td>11.</td><td>maldinna mummi</td><td>= 1 of the toes added on (to the 10 fingers).</td></tr> +<tr><td>12.</td><td>bular dinna mummi</td><td>= 2 of the toes added on.</td></tr> +<tr><td>13.</td><td>guliba dinna mummi</td><td>= 3 of the toes added on.</td></tr> +<tr><td>14.</td><td>bular bular dinna mummi</td><td>= 4 of the toes added on.</td></tr> +<tr><td>15.</td><td>mulanba dinna</td><td>= 5 of the toes added on.</td></tr> +<tr><td>16.</td><td>mal dinna mulanbu</td><td>= 1 and 5 toes.</td></tr> +<tr><td>17.</td><td>bular dinna mulanbu</td><td>= 2 and 5 toes.</td></tr> +<tr><td>18.</td><td>guliba dinna mulanbu</td><td>= 3 and 5 toes.</td></tr> +<tr><td>19.</td><td>bular bular dinna mulanbu</td><td>= 4 and 5 toes.</td></tr> +<tr><td>20.</td><td>bularin dinna</td><td>= belonging to the 2 feet.</td></tr> +</table> + + +<p>As has already been stated, there is good ground for +believing that this system was originally as limited as +those obtained from other Australian tribes, and that +its extension from 4, or perhaps from 5 onward, is of +comparatively recent date.</p> + +<p>A somewhat peculiar numeral nomenclature is found +in the language of the Klamath Indians of Oregon. +The first ten words in the Klamath scale are:<a id="FNA-81" href="#FN-81"><sup>81</sup></a></p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">nash, or nas.</td></tr> +<tr><td>2.</td><td>lap</td><td>= hand.</td></tr> +<tr><td>3.</td><td colspan="2">ndan.</td></tr> +<tr><td>4.</td><td>vunep</td><td>= hand up.</td></tr> +<tr><td>5.</td><td>tunep</td><td>= hand away.</td></tr> +<tr><td>6.</td><td>nadshkshapta</td><td>= 1 I have bent over.</td></tr> +<tr><td>7.</td><td>lapkshapta</td><td>= 2 I have bent over.</td></tr> +<tr><td>8.</td><td>ndankshapta</td><td>= 3 I have bent over.</td></tr> +<tr><td>9.</td><td>nadshskeksh</td><td>= 1 left over.</td></tr> +<tr><td>10.</td><td>taunep</td><td>= hand hand?</td></tr> +</table> + + +<p><a class="pgnum" id="page-59" title="- 59 -"></a>In describing this system Mr. Gatschet says: “If +the origin of the Klamath numerals is thus correctly +traced, their inventors must have counted only the +four long fingers without the thumb, and 5 was counted +while saying <i>hand away! hand off!</i> The ‘four,’ or <i>hand +high! hand up!</i> intimates that the hand was held up +high after counting its four digits; and some term +expressing this gesture was, in the case of <i>nine</i>, substituted +by ‘one left over’ … which means to say, +‘only one is left until all the fingers are counted.’” It +will be observed that the Klamath introduces not only +the ordinary finger manipulation, but a gesture of the +entire hand as well. It is a common thing to find +something of the kind to indicate the completion of 5 +or 10, and in one or two instances it has already been +alluded to. Sometimes one or both of the closed fists +are held up; sometimes the open hand, with all the +fingers extended, is used; and sometimes an entirely +independent gesture is introduced. These are, in general, +of no special importance; but one custom in vogue +among some of the prairie tribes of Indians, to which +my attention was called by Dr. J. Owen Dorsey,<a id="FNA-82" href="#FN-82"><sup>82</sup></a> +should be mentioned. It is a gesture which signifies +multiplication, and is performed by throwing the hand +to the left. Thus, after counting 5, a wave of the +hand to the left means 50. As multiplication is rather +unusual among savage tribes, this is noteworthy, and +would seem to indicate on the part of the Indian a +<a class="pgnum" id="page-60" title="- 60 -"></a>higher degree of intelligence than is ordinarily possessed +by uncivilized races.</p> + +<p>In the numeral scale as we possess it in English, we +find it necessary to retain the name of the last unit of +each kind used, in order to describe definitely any +numeral employed. Thus, fifteen, one hundred forty-two, +six thousand seven hundred twenty-seven, give in +full detail the numbers they are intended to describe. +In primitive scales this is not always considered necessary; +thus, the Zamucos express their teens without +using their word for 10 at all. They say simply, 1 on +the foot, 2 on the foot, etc. Corresponding abbreviations +are often met; so often, indeed, that no further +mention of them is needed. They mark one extreme, +the extreme of brevity, found in the savage method of +building up hand, foot, and finger names for numerals; +while the Zuñi scale marks the extreme of prolixity +in the formation of such words. A somewhat ruder +composition than any yet noticed is shown in the +numerals of the Vilelo scale,<a id="FNA-83" href="#FN-83"><sup>83</sup></a> which are:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">agit, or yaagit.</td></tr> +<tr><td>2.</td><td colspan="2">uke.</td></tr> +<tr><td>3.</td><td colspan="2">nipetuei.</td></tr> +<tr><td>4.</td><td colspan="2">yepkatalet.</td></tr> +<tr><td>5.</td><td>isig-nisle-yaagit</td><td>= hand fingers 1.</td></tr> +<tr><td>6.</td><td>isig-teet-yaagit</td><td>= hand with 1.</td></tr> +<tr><td>7.</td><td>isig-teet-uke</td><td>= hand with 2.</td></tr> +<tr><td>8.</td><td>isig-teet-nipetuei</td><td>= hand with 3.</td></tr> +<tr><td>9.</td><td>isig-teet-yepkatalet</td><td>= hand with 4.</td></tr> +<tr><td>10.</td><td>isig-uke-nisle</td><td>= second hand fingers (lit. hand-two-fingers).</td></tr> +<tr><td>11.</td><td>isig-uke-nisle-teet-yaagit</td><td>= second hand fingers with 1.</td></tr> +<tr><td>20.</td><td>isig-ape-nisle-lauel</td><td>= hand foot fingers all.</td></tr> +</table> + +<p><a class="pgnum" id="page-61" title="- 61 -"></a>In the examples thus far given, it will be noticed +that the actual names of individual fingers do not +appear. In general, such words as thumb, forefinger, +little finger, are not found, but rather the hand-1, 1 on +the next, or 1 over and above, which we have already +seen, are the type forms for which we are to look. +Individual finger names do occur, however, as in the +scale of the Hudson's Bay Eskimos,<a id="FNA-84" href="#FN-84"><sup>84</sup></a> where the three following +words are used both as numerals and as finger +names:</p> + +<table class="scale"> +<tr><td>8.</td><td>kittukleemoot</td><td>= middle finger.</td></tr> +<tr><td>9.</td><td>mikkeelukkamoot</td><td>= fourth finger.</td></tr> +<tr><td>10.</td><td>eerkitkoka</td><td>= little finger.</td></tr> +</table> + +<p>Words of similar origin are found in the original +Jiviro scale,<a id="FNA-85" href="#FN-85"><sup>85</sup></a> where the native numerals are:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">ala.</td></tr> +<tr><td>2.</td><td colspan="2">catu.</td></tr> +<tr><td>3.</td><td colspan="2">cala.</td></tr> +<tr><td>4.</td><td colspan="2">encatu.</td></tr> +<tr><td>5.</td><td>alacötegladu</td><td>= 1 hand.</td></tr> +<tr><td>6.</td><td>intimutu</td><td>= thumb (of second hand).</td></tr> +<tr><td>7.</td><td>tannituna</td><td>= index finger.</td></tr> +<tr><td>8.</td><td>tannituna cabiasu</td><td>= the finger next the index finger.</td></tr> +<tr><td>9.</td><td>bitin ötegla cabiasu</td><td>= hand next to complete.</td></tr> +<tr><td>10.</td><td>catögladu</td><td>= 2 hands.</td></tr> +</table> + +<p>As if to emphasize the rarity of this method of forming +numerals, the Jiviros afterward discarded the last +five of the above scale, replacing them by words borrowed +from the Quichuas, or ancient Peruvians. The +same process may have been followed by other tribes, +<a class="pgnum" id="page-62" title="- 62 -"></a>and in this way numerals which were originally digital +may have disappeared. But we have no evidence that +this has ever happened in any extensive manner. We +are, rather, impelled to accept the occasional numerals +of this class as exceptions to the general rule, until we +have at our disposal further evidence of an exact and +critical nature, which would cause us to modify this +opinion. An elaborate philological study by Dr. J. H. +Trumbull<a id="FNA-86" href="#FN-86"><sup>86</sup></a> of the numerals used by many of the North +American Indian tribes reveals the presence in the +languages of these tribes of a few, but only a few, +finger names which are used without change as numeral +expressions also. Sometimes the finger gives a name +not its own to the numeral with which it is associated +in counting—as in the Chippeway dialect, which has +<i>nawi-nindj</i>, middle of the hand, and <i>nisswi</i>, 3; and the +Cheyenne, where <i>notoyos</i>, middle finger, and <i>na-nohhtu</i>, +8, are closely related. In other parts of the world +isolated examples of the transference of finger names +to numerals are also found. Of these a well-known +example is furnished by the Zulu numerals, where +“<i>tatisitupa</i>, taking the thumb, becomes a numeral for +six. Then the verb <i>komba</i>, to point, indicating the +forefinger, or ‘pointer,’ makes the next numeral, seven. +Thus, answering the question, ‘How much did your +master give you?’ a Zulu would say, ‘<i>U kombile</i>,’ ‘He +pointed with his forefinger,’ <i>i.e.</i> ‘He gave me seven’; +and this curious way of using the numeral verb is also +<a class="pgnum" id="page-63" title="- 63 -"></a>shown in such an example as ‘<i>amahasi akombile</i>,’ ‘the +horses have pointed,’ <i>i.e.</i> ‘there were seven of them.’ +In like manner, <i>Kijangalobili</i>, ‘keep back two fingers,’ +<i>i.e.</i> eight, and <i>Kijangalolunje</i>, ‘keep back one finger,’ +<i>i.e.</i> nine, lead on to <i>kumi</i>, ten.”<a id="FNA-87" href="#FN-87"><sup>87</sup></a></p> + +<p>Returning for a moment to the consideration of number +systems in the formation of which the influence of the +hand has been paramount, we find still further variations +of the method already noticed of constructing names for +the fives, tens, and twenties, as well as for the intermediate +numbers. Instead of the simple words “hand,” +“foot,” etc., we not infrequently meet with some paraphrase +for one or for all these terms, the derivation of +which is unmistakable. The Nengones,<a id="FNA-88" href="#FN-88"><sup>88</sup></a> an island tribe +of the Indian Ocean, though using the word “man” for +20, do not employ explicit hand or foot words, but count</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">sa.</td></tr> +<tr><td>2.</td><td colspan="2">rewe.</td></tr> +<tr><td>3.</td><td colspan="2">tini.</td></tr> +<tr><td>4.</td><td colspan="2">etse.</td></tr> +<tr><td>5.</td><td>se dono</td><td>= the end (of the first hand).</td></tr> +<tr><td>6.</td><td>dono ne sa</td><td>= end and 1.</td></tr> +<tr><td>7.</td><td>dono ne rewe</td><td>= end and 2.</td></tr> +<tr><td>8.</td><td>dono ne tini</td><td>= end and 3.</td></tr> +<tr><td>9.</td><td>dono ne etse</td><td>= end and 4.</td></tr> +<tr><td>10.</td><td>rewe tubenine</td><td>= 2 series (of fingers).</td></tr> +<tr><td>11.</td><td>rewe tubenine ne sa re tsemene</td><td>= 2 series and 1 on the next?</td></tr> +<tr><td>20.</td><td>sa re nome</td><td>= 1 man.</td></tr> +<tr><td>30.</td><td>sa re nome ne rewe tubenine</td><td>= 1 man and 2 series.</td></tr> +<tr><td>40.</td><td>rewe ne nome</td><td>= 2 men.</td></tr> +</table> + +<p>Examples like the above are not infrequent. The +Aztecs used for 10 the word <i>matlactli</i>, hand-half, <i>i.e.</i> the +<a class="pgnum" id="page-64" title="- 64 -"></a>hand half of a man, and for 20 <i>cempoalli</i>, one counting.<a id="FNA-89" href="#FN-89"><sup>89</sup></a> +The Point Barrow Eskimos call 10 <i>kodlin</i>, the upper part, +<i>i.e.</i> of a man. One of the Ewe dialects of Western +Africa<a id="FNA-90" href="#FN-90"><sup>90</sup></a> has <i>ewo</i>, done, for 10; while, curiously enough, +9, <i>asieke</i>, is a digital word, meaning “to part (from) the +hand.”</p> + +<p>In numerous instances also some characteristic word +not of hand derivation is found, like the Yoruba <i>ogodzi</i>, +string, which becomes a numeral for 40, because 40 +cowries made a “string”; and the Maori <i>tekau</i>, bunch, +which signifies 10. The origin of this seems to have +been the custom of counting yams and fish by “bunches” +of ten each.<a id="FNA-91" href="#FN-91"><sup>91</sup></a></p> + +<p>Another method of forming numeral words above 5 +or 10 is found in the presence of such expressions as +second 1, second 2, etc. In languages of rude construction +and incomplete development the simple numeral +scale is often found to end with 5, and all succeeding +numerals to be formed from the first 5. The progression +from that point may be 5-1, 5-2, etc., as in the +numerous quinary scales to be noticed later, or it may +be second 1, second 2, etc., as in the Niam Niam dialect +of Central Africa, where the scale is<a id="FNA-92" href="#FN-92"><sup>92</sup></a></p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">sa.</td></tr> +<tr><td>2.</td><td colspan="2">uwi.</td></tr> +<tr><td>3.</td><td colspan="2">biata.</td></tr> +<tr><td>4.</td><td colspan="2">biama.</td></tr> +<tr><td>5.</td><td colspan="2">biswi.</td></tr> +<tr><td>6.</td><td>batissa</td><td>= 2d 1.</td></tr> +<tr><td><a class="pgnum" id="page-65" title="- 65 -"></a>7.</td><td>batiwwi</td><td>= 2d 2.</td></tr> +<tr><td>8.</td><td>batti-biata</td><td>= 2d 3.</td></tr> +<tr><td>9.</td><td>batti-biama</td><td>= 2d 4.</td></tr> +<tr><td>10.</td><td>bauwé</td><td>= 2d 5.</td></tr> +</table> + +<p>That this method of progression is not confined to the +least developed languages, however, is shown by a most +cursory examination of the numerals of our American +Indian tribes, where numeral formation like that exhibited +above is exceedingly common. In the Kootenay +dialect,<a id="FNA-93" href="#FN-93"><sup>93</sup></a> of British Columbia, <i>qaetsa</i>, 4, and <i>wo-qaetsa,</i> 8, +are obviously related, the latter word probably meaning +a second 4. Most of the native languages of British +Columbia form their words for 7 and 8 from those +which signify 2 and 3; as, for example, the Heiltsuk,<a id="FNA-94" href="#FN-94"><sup>94</sup></a> +which shows in the following words a most obvious +correspondence:</p> + +<table class="scale"> +<tr><td>2.</td><td>matl.</td><td class="number">7.</td><td>matlaaus.</td></tr> +<tr><td>3.</td><td>yutq.</td><td class="number">8.</td><td>yutquaus.</td></tr> +</table> + +<p>In the Choctaw language<a id="FNA-95" href="#FN-95"><sup>95</sup></a> the relation between 2 and +7, and 3 and 8, is no less clear. Here the words are:</p> + +<table class="scale"> +<tr><td>2.</td><td>tuklo.</td><td class="number">7.</td><td>untuklo.</td></tr> +<tr><td>3.</td><td>tuchina.</td><td class="number">8.</td><td>untuchina.</td></tr> +</table> + +<p>The Nez Percés<a id="FNA-96" href="#FN-96"><sup>96</sup></a> repeat the first three words of their +scale in their 6, 7, and 8 respectively, as a comparison of +these numerals will show.</p> + +<table class="scale"> +<tr><td><a class="pgnum" id="page-66" title="- 66 -"></a>1.</td><td>naks.</td><td class="number">6.</td><td>oilaks.</td></tr> +<tr><td>2.</td><td>lapit.</td><td class="number">7.</td><td>oinapt.</td></tr> +<tr><td>3.</td><td>mitat.</td><td class="number">8.</td><td>oimatat.</td></tr> +</table> + +<p>In all these cases the essential point of the method +is contained in the repetition, in one way or another, +of the numerals of the second quinate, without the use +with each one of the word for 5. This may make 6, +7, 8, and 9 appear as second 1, second 2, etc., or another +1, another 2, etc.; or, more simply still, as 1 more, 2 +more, etc. It is the method which was briefly discussed +in the early part of the present chapter, and is by no +means uncommon. In a decimal scale this repetition +would begin with 11 instead of 6; as in the system found +in use in Tagala and Pampanaga, two of the Philippine +Islands, where, for example, 11, 12, and 13 are:<a id="FNA-97" href="#FN-97"><sup>97</sup></a></p> + +<table class="scale"> +<tr><td>11.</td><td>labi-n-isa</td><td>= over 1.</td></tr> +<tr><td>12.</td><td>labi-n-dalaua</td><td>= over 2.</td></tr> +<tr><td>13.</td><td>labi-n-tatlo</td><td>= over 3.</td></tr> +</table> + +<p>A precisely similar method of numeral building is used +by some of our Western Indian tribes. Selecting a few +of the Assiniboine numerals<a id="FNA-98" href="#FN-98"><sup>98</sup></a> as an illustration, we have</p> + +<table class="scale"> +<tr><td>11.</td><td>ak kai washe</td><td>= more 1.</td></tr> +<tr><td>12.</td><td>ak kai noom pah</td><td>= more 2.</td></tr> +<tr><td>13.</td><td>ak kai yam me nee</td><td>= more 3.</td></tr> +<tr><td>14.</td><td>ak kai to pah</td><td>= more 4.</td></tr> +<tr><td>15.</td><td>ak kai zap tah</td><td>= more 5.</td></tr> +<tr><td>16.</td><td>ak kai shak pah</td><td>= more 6, etc.</td></tr> +</table> + +<p>A still more primitive structure is shown in the +numerals of the Mboushas<a id="FNA-99" href="#FN-99"><sup>99</sup></a> of Equatorial Africa. Instead +of using 5-1, 5-2, 5-3, 5-4, or 2d 1, 2d 2, 2d 3, +<a class="pgnum" id="page-67" title="- 67 -"></a>2d 4, in forming their numerals from 6 to 9, they proceed +in the following remarkable and, at first thought, +inexplicable manner to form their compound numerals:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">ivoco.</td></tr> +<tr><td>2.</td><td colspan="2">beba.</td></tr> +<tr><td>3.</td><td colspan="2">belalo.</td></tr> +<tr><td>4.</td><td colspan="2">benai.</td></tr> +<tr><td>5.</td><td colspan="2">betano.</td></tr> +<tr><td>6.</td><td>ivoco beba</td><td>= 1-2.</td></tr> +<tr><td>7.</td><td>ivoco belalo</td><td>= 1-3.</td></tr> +<tr><td>8.</td><td>ivoco benai</td><td>= 1-4.</td></tr> +<tr><td>9.</td><td>ivoco betano</td><td>= 1-5.</td></tr> +<tr><td>10.</td><td colspan="2">dioum.</td></tr> +</table> + +<p>No explanation is given by Mr. du Chaillu for such +an apparently incomprehensible form of expression as, +for example, 1-3, for 7. Some peculiar finger pantomime +may accompany the counting, which, were it +known, would enlighten us on the Mbousha's method +of arriving at so anomalous a scale. Mere repetition +in the second quinate of the words used in the first +might readily be explained by supposing the use of fingers +absolutely indispensable as an aid to counting, and +that a certain word would have one meaning when associated +with a certain finger of the left hand, and another +meaning when associated with one of the fingers of the +right. Such scales are, if the following are correct, +actually in existence among the islands of the Pacific.</p> + +<div class="lefthalf"> +<table class="scale"> +<caption>Balad.<a id="FNA-100" href="#FN-100"><sup>100</sup></a></caption> +<tr><td>1.</td><td>parai.</td></tr> +<tr><td>2.</td><td>paroo.</td></tr> +<tr><td>3.</td><td>pargen.</td></tr> +<tr><td>4.</td><td>parbai.</td></tr> +<tr><td>5.</td><td>panim.</td></tr> + +<tr><td>6.</td><td>parai.</td></tr> +<tr><td>7.</td><td>paroo.</td></tr> +<tr><td>8.</td><td>pargen.</td></tr> +<tr><td>9.</td><td>parbai.</td></tr> +<tr><td>10.</td><td>panim.</td></tr> +</table> +</div> +<div class="righthalf"> +<table class="scale"> +<caption>Uea.<a href="#FN-100"><sup>100</sup></a></caption> +<tr><td>1.</td><td>tahi.</td></tr> +<tr><td>2.</td><td>lua.</td></tr> +<tr><td>3.</td><td>tolu.</td></tr> +<tr><td>4.</td><td>fa.</td></tr> +<tr><td>5.</td><td>lima.</td></tr> +<tr><td>6.</td><td>tahi.</td></tr> +<tr><td>7.</td><td>lua.</td></tr> +<tr><td>8.</td><td>tolu.</td></tr> +<tr><td>9.</td><td>fa.</td></tr> +<tr><td>10.</td><td>lima.</td></tr> +</table> +</div> + +<p><a class="pgnum" id="page-68" title="- 68 -"></a>Such examples are, I believe, entirely unique among +primitive number systems.</p> + +<p>In numeral scales where the formative process has +been of the general nature just exhibited, irregularities +of various kinds are of frequent occurrence. Hand +numerals may appear, and then suddenly disappear, +just where we should look for them with the greatest +degree of certainty. In the Ende,<a id="FNA-101" href="#FN-101"><sup>101</sup></a> a dialect of the +Flores Islands, 5, 6, and 7 are of hand formation, while +8 and 9 are of entirely different origin, as the scale +shows.</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">sa.</td></tr> +<tr><td>2.</td><td colspan="2">zua.</td></tr> +<tr><td>3.</td><td colspan="2">telu.</td></tr> +<tr><td>4.</td><td colspan="2">wutu.</td></tr> +<tr><td>5.</td><td colspan="2">lima</td></tr> +<tr><td>6.</td><td>lima sa</td><td>= hand 1.</td></tr> +<tr><td>7.</td><td>lima zua</td><td>= hand 2.</td></tr> +<tr><td>8.</td><td>rua butu</td><td>= 2 × 4.</td></tr> +<tr><td>9.</td><td>trasa</td><td>= 10 − 1?</td></tr> +<tr><td>10.</td><td colspan="2">sabulu.</td></tr> +</table> + +<p>One special point to be noticed in this scale is the +irregularity that prevails between 7, 8, 9. The formation +of 7 is of the most ordinary kind; 8 is 2 fours—common +enough duplication; while 9 appears to be +10 − 1. All of these modes of compounding are, in +their own way, regular; but the irregularity consists in +using all three of them in connective numerals in the +same system. But, odd as this jumble seems, it is more +than matched by that found in the scale of the Karankawa +Indians,<a id="FNA-102" href="#FN-102"><sup>102</sup></a> an extinct tribe formerly inhabiting the +<a class="pgnum" id="page-69" title="- 69 -"></a>coast region of Texas. The first ten numerals of this +singular array are:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">natsa.</td></tr> +<tr><td>2.</td><td colspan="2">haikia.</td></tr> +<tr><td>3.</td><td colspan="2">kachayi.</td></tr> +<tr><td>4.</td><td>hayo hakn</td><td>= 2 × 2.</td></tr> +<tr><td>5.</td><td>natsa behema</td><td>= 1 father, <i>i.e.</i> of the fingers.</td></tr> +<tr><td>6.</td><td>hayo haikia</td><td>= 3 × 2?</td></tr> +<tr><td>7.</td><td>haikia natsa</td><td>= 2 + 5?</td></tr> +<tr><td>8.</td><td>haikia behema</td><td>= 2 fathers?</td></tr> +<tr><td>9.</td><td>haikia doatn</td><td>= 2d from 10?</td></tr> +<tr><td>10.</td><td colspan="2">doatn habe.</td></tr> +</table> + +<p>Systems like the above, where chaos instead of order +seems to be the ruling principle, are of occasional +occurrence, but they are decidedly the exception.</p> + +<p>In some of the cases that have been adduced for illustration +it is to be noticed that the process of combination +begins with 7 instead of with 6. Among others, +the scale of the Pigmies of Central Africa<a id="FNA-103" href="#FN-103"><sup>103</sup></a> and that +of the Mosquitos<a id="FNA-104" href="#FN-104"><sup>104</sup></a> of Central America show this tendency. +In the Pigmy scale the words for 1 and 6 +are so closely akin that one cannot resist the impression +that 6 was to them a new 1, and was thus named.</p> + +<table class="scale"> +<thead><tr><th></th><th class="caption">Mosquito.</th><th class="caption">Pigmy.</th></tr></thead> + +<tr><td>1.</td><td>kumi.</td><td>ujju.</td></tr> +<tr><td>2.</td><td>wal.</td><td>ibari.</td></tr> +<tr><td>3.</td><td>niupa.</td><td>ikaro.</td></tr> +<tr><td>4.</td><td>wal-wal = 2-2.</td><td>ikwanganya.</td></tr> +<tr><td>5.</td><td>mata-sip = fingers of 1 hand.</td><td>bumuti.</td></tr> +<tr><td>6.</td><td>matlalkabe.</td><td>ijju.</td></tr> +<tr><td>7.</td><td>matlalkabe pura kumi = 6 and 1.</td><td>bumutti-na-ibali = 5 and 2.</td></tr> +<tr><td><a class="pgnum" id="page-70" title="- 70 -"></a>8.</td><td>matlalkabe pura wal = 6 and 2.</td><td>bumutti-na-ikaro = 5 and 3.</td></tr> +<tr><td>9.</td><td>matlalkabe pura niupa = 6 and 3.</td><td>bumutti-na-ikwanganya = 5 and 4.</td></tr> +<tr><td>10.</td><td>mata wal sip = fingers of 2 hands.</td><td>mabo = half man.</td></tr> +</table> + + + +<p>The Mosquito scale is quite exceptional in forming +7, 8, and 9 from 6, instead of from 5. The usual +method, where combinations appear between 6 and 10, +is exhibited by the Pigmy scale. Still another species +of numeral form, quite different from any that have +already been noticed, is found in the Yoruba<a id="FNA-105" href="#FN-105"><sup>105</sup></a> scale, +which is in many respects one of the most peculiar in +existence. Here the words for 11, 12, etc., are formed +by adding the suffix <i>-la</i>, great, to the words for 1, 2, +etc., thus:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">eni, or okan.</td></tr> +<tr><td>2.</td><td colspan="2">edzi.</td></tr> +<tr><td>3.</td><td colspan="2">eta.</td></tr> +<tr><td>4.</td><td colspan="2">erin.</td></tr> +<tr><td>5.</td><td colspan="2">arun.</td></tr> +<tr><td>6.</td><td colspan="2">efa.</td></tr> +<tr><td>7.</td><td colspan="2">edze.</td></tr> +<tr><td>8.</td><td colspan="2">edzo.</td></tr> +<tr><td>9.</td><td colspan="2">esan.</td></tr> +<tr><td>10.</td><td colspan="2">ewa.</td></tr> +<tr><td>11.</td><td>okanla</td><td>= great 1.</td></tr> +<tr><td>12.</td><td>edzila</td><td>= great 2.</td></tr> +<tr><td>13.</td><td>etala</td><td>= great 3.</td></tr> +<tr><td>14.</td><td>erinla</td><td>= great 4, etc.</td></tr> +<tr><td>40.</td><td>ogodzi</td><td>= string.</td></tr> +<tr><td>200.</td><td>igba</td><td>= heap.</td></tr> +</table> + +<p>The word for 40 was adopted because cowrie shells, +which are used for counting, were strung by forties; +and <i>igba</i>, 200, because a heap of 200 shells was five +strings, and thus formed a convenient higher unit for +<a class="pgnum" id="page-71" title="- 71 -"></a>reckoning. Proceeding in this curious manner,<a id="FNA-106" href="#FN-106"><sup>106</sup></a> they +called 50 strings 1 <i>afo</i> or head; and to illustrate their +singular mode of reckoning—the king of the Dahomans, +having made war on the Yorubans, and attacked +their army, was repulsed and defeated with a loss of +“two heads, twenty strings, and twenty cowries” of +men, or 4820.</p> + +<p>The number scale of the Abipones,<a id="FNA-107" href="#FN-107"><sup>107</sup></a> one of the low +tribes of the Paraguay region, contains two genuine +curiosities, and by reason of those it deserves a place +among any collection of numeral scales designed to +exhibit the formation of this class of words. It is:</p> + +<table class="scale"> +<tr><td>1.</td><td>initara</td><td>= 1 alone.</td></tr> +<tr><td>2.</td><td colspan="2">inoaka.</td></tr> +<tr><td>3.</td><td>inoaka yekaini</td><td>= 2 and 1.</td></tr> +<tr><td>4.</td><td>geyenknate</td><td>= toes of an ostrich.</td></tr> +<tr><td>5.</td><td>neenhalek</td><td>= a five coloured, spotted hide,</td></tr> +<tr><td></td><td class="alt">or hanambegen</td><td>= fingers of 1 hand.</td></tr> +<tr><td>10.</td><td>lanamrihegem</td><td>= fingers of both hands.</td></tr> +<tr><td>20.</td><td colspan="2">lanamrihegem cat gracherhaka anamichirihegem = fingers of both hands together with toes of both feet.</td></tr> +</table> + +<p>That the number sense of the Abipones is but little, +if at all, above that of the native Australian tribes, is +shown by their expressing 3 by the combination 2 +and 1. This limitation, as we have already seen, is +shared by the Botocudos, the Chiquitos, and many of +<a class="pgnum" id="page-72" title="- 72 -"></a>the other native races of South America. But the +Abipones, in seeking for words with which to enable +themselves to pass beyond the limit 3, invented the +singular terms just given for 4 and 5. The ostrich, +having three toes in front and one behind on each foot +presented them with a living example of 3 + 1; hence +“toes of an ostrich” became their numeral for 4. Similarly, +the number of colours in a certain hide being five, +the name for that hide was adopted as their next +numeral. At this point they began to resort to digital +numeration also; and any higher number is expressed +by that method.</p> + +<p>In the sense in which the word is defined by mathematicians, +<i>number</i> is a pure, abstract concept. But a +moment's reflection will show that, as it originates +among savage races, number is, and from the limitations +of their intellect must be, entirely concrete. An abstract +conception is something quite foreign to the essentially +primitive mind, as missionaries and explorers have found +to their chagrin. The savage can form no mental concept +of what civilized man means by such a word as +“soul”; nor would his idea of the abstract number 5 be +much clearer. When he says <i>five</i>, he uses, in many cases +at least, the same word that serves him when he wishes +to say <i>hand</i>; and his mental concept when he says <i>five</i> +is of a hand. The concrete idea of a closed fist or an +open hand with outstretched fingers, is what is upper-most +in his mind. He knows no more and cares no +more about the pure number 5 than he does about the +<a class="pgnum" id="page-73" title="- 73 -"></a>law of the conservation of energy. He sees in his +mental picture only the real, material image, and his +only comprehension of the number is, “these objects +are as many as the fingers on my hand.” Then, in +the lapse of the long interval of centuries which intervene +between lowest barbarism and highest civilization, +the abstract and the concrete become slowly dissociated, +the one from the other. First the actual hand picture +fades away, and the number is recognized without the +original assistance furnished by the derivation of the +word. But the number is still for a long time a certain +number <i>of objects</i>, and not an independent concept. +It is only when the savage ceases to be wholly an +animal, and becomes a thinking human being, that +number in the abstract can come within the grasp of +his mind. It is at this point that mere reckoning +ceases, and arithmetic begins.</p> + + + + + +<h2><a class="pgnum" id="page-74" title="- 74 -"></a>Chapter IV.</h2> +<h3>The Origin of Number Words.<br/> +(<i>Continued</i>.)</h3> + + +<p><span class="initial">By</span> the slow, and often painful, process incident to +the extension and development of any mental conception +in a mind wholly unused to abstractions, the +savage gropes his way onward in his counting from 1, +or more probably from 2, to the various higher numbers +required to form his scale. The perception of +unity offers no difficulty to his mind, though he is +conscious at first of the object itself rather than of any +idea of number associated with it. The concept of +duality, also, is grasped with perfect readiness. This +concept is, in its simplest form, presented to the mind +as soon as the individual distinguishes himself from +another person, though the idea is still essentially +concrete. Perhaps the first glimmering of any real +number thought in connection with 2 comes when the +savage contrasts one single object with another—or, +in other words, when he first recognizes the <i>pair</i>. At +first the individuals composing the pair are simply +“this one,” and “that one,” or “this and that”; and +his number system now halts for a time at the stage +<a class="pgnum" id="page-75" title="- 75 -"></a>when he can, rudely enough it may be, count 1, 2, +many. There are certain cases where the forms of 1 +and 2 are so similar <del title="obviously should read ‘that’">than</del><ins title="than for that: probable printer's error">that</ins> one may readily imagine +that these numbers really were “this” and “that” in +the savage's original conception of them; and the same +likeness also occurs in the words for 3 and 4, which +may readily enough have been a second “this” and a +second “that.” In the Lushu tongue the words for 1 +and 2 are <i>tizi</i> and <i>tazi</i> respectively. In Koriak we find +<i>ngroka</i>, 3, and <i>ngraka</i>, 4; in Kolyma, <i>niyokh</i>, 3, and +<i>niyakh</i>, 4; and in Kamtschatkan, <i>tsuk</i>, 3, and <i>tsaak</i>, 4.<a id="FNA-108" href="#FN-108"><sup>108</sup></a> +Sometimes, as in the case of the Australian races, the +entire extent of the count is carried through by means +of pairs. But the natural theory one would form is, +that 2 is the halting place for a very long time; that +up to this point the fingers may or may not have +been used—probably not; and that when the next +start is made, and 3, 4, 5, and so on are counted, the +fingers first come into requisition. If the grammatical +structure of the earlier languages of the world's history +is examined, the student is struck with the prevalence +of the dual number in them—something which +tends to disappear as language undergoes extended development. +The dual number points unequivocally to +the time when 1 and 2 were <i>the</i> numbers at mankind's +disposal; to the time when his three numeral concepts, +1, 2, many, each demanded distinct expression. With +increasing knowledge the necessity for this differentiatuin +<a class="pgnum" id="page-76" title="- 76 -"></a>would pass away, and but two numbers, singular +and plural, would remain. Incidentally it is to be +noticed that the Indo-European words for 3—<i>three</i>, +<i>trois</i>, <i>drei</i>, <i>tres</i>, <i>tri,</i> etc., have the same root as the +Latin <i>trans</i>, beyond, and give us a hint of the time +when our Aryan ancestors counted in the manner I +have just described.</p> + +<p>The first real difficulty which the savage experiences +in counting, the difficulty which comes when he attempts +to pass beyond 2, and to count 3, 4, and 5, is of course +but slight; and these numbers are commonly used and +readily understood by almost all tribes, no matter how +deeply sunk in barbarism we find them. But the instances +that have already been cited must not be forgotten. +The Chiquitos do not, in their primitive state, +properly count at all; the Andamans, the Veddas, and +many of the Australian tribes have no numerals higher +than 2; others of the Australians and many of the South +Americans stop with 3 or 4; and tribes which make 5 +their limit are still more numerous. Hence it is safe to +assert that even this insignificant number is not always +reached with perfect ease. Beyond 5 primitive man +often proceeds with the greatest difficulty. Most savages, +even those of the tribes just mentioned, can really +count above here, even though they have no words with +which to express their thought. But they do it with +reluctance, and as they go on they quickly lose all sense +of accuracy. This has already been commented on, but +to emphasize it afresh the well-known example given by +<a class="pgnum" id="page-77" title="- 77 -"></a>Mr. Oldfield from his own experience among the Watchandies +may be quoted.<a id="FNA-109" href="#FN-109"><sup>109</sup></a> “I once wished to ascertain the +exact number of natives who had been slain on a certain +occasion. The individual of whom I made the inquiry +began to think over the names … assigning one of +his fingers to each, and it was not until after many failures, +and consequent fresh starts, that he was able to +express so high a number, which he at length did by +holding up his hand three times, thus giving me to understand +that fifteen was the answer to this most difficult +arithmetical question.” This meagreness of knowledge +in all things pertaining to numbers is often found to be +sharply emphasized in the names adopted by savages for +their numeral words. While discussing in a previous +chapter the limits of number systems, we found many +instances where anything above 2 or 3 was designated +by some one of the comprehensive terms <i>much</i>, <i>many</i>, +<i>very many</i>; these words, or such equivalents as <i>lot</i>, <i>heap</i>, +or <i>plenty</i>, serving as an aid to the finger pantomime +necessary to indicate numbers for which they have no +real names. The low degree of intelligence and civilization +revealed by such words is brought quite as +sharply into prominence by the word occasionally found +for 5. Whenever the fingers and hands are used at all, +it would seem natural to expect for 5 some general +expression signifying <i>hand</i>, for 10 <i>both hands</i>, and for +20 <i>man</i>. Such is, as we have already seen, the ordinary +method of progression, but it is not universal. A drop +<a class="pgnum" id="page-78" title="- 78 -"></a>in the scale of civilization takes us to a point where 10, +instead of 20, becomes the whole man. The Kusaies,<a id="FNA-110" href="#FN-110"><sup>110</sup></a> +of Strong's Island, call 10 <i>sie-nul</i>, 1 man, 30 <i>tol-nul</i>, 3 +men, 40 <i>a naul</i>, 4 men, etc.; and the Ku-Mbutti<a id="FNA-111" href="#FN-111"><sup>111</sup></a> of +central Africa have <i>mukko</i>, 10, and <i>moku</i>, man. If 10 +is to be expressed by reference to the man, instead of +his hands, it might appear more natural to employ some +such expression as that adopted by the African Pigmies,<a id="FNA-112" href="#FN-112"><sup>112</sup></a> +who call 10 <i>mabo</i>, and man <i>mabo-mabo</i>. With them, then, +10 is perhaps “half a man,” as it actually is among the +Towkas of South America; and we have already seen +that with the Aztecs it was <i>matlactli</i>, the “hand half” +of a man.<a id="FNA-113" href="#FN-113"><sup>113</sup></a> The same idea crops out in the expression +used by the Nicobar Islanders for 30—<i>heam-umdjome +ruktei</i>, 1 man (and a) half.<a id="FNA-114" href="#FN-114"><sup>114</sup></a> Such nomenclature is +entirely natural, and it accords with the analogy offered +by other words of frequent occurrence in the numeral +scales of savage races. Still, to find 10 expressed by the +term <i>man</i> always conveys an impression of mental poverty; +though it may, of course, be urged that this might +arise from the fact that some races never use the toes +in counting, but go over the fingers again, or perhaps +bring into requisition the fingers of a second man to +express the second 10. It is not safe to postulate an +extremely low degree of civilization from the presence +<a class="pgnum" id="page-79" title="- 79 -"></a>of certain peculiarities of numeral formation. Only the +most general statements can be ventured on, and these +are always subject to modification through some circumstance +connected with environment, mode of living, or +intercourse with other tribes. Two South American +races may be cited, which seem in this respect to give +unmistakable evidence of being sunk in deepest barbarism. +These are the Juri and the Cayriri, who use the +same word for man and for 5. The former express 5 +by <i>ghomen apa</i>, 1 man,<a id="FNA-115" href="#FN-115"><sup>115</sup></a> and the latter by <i>ibicho</i>, person.<a id="FNA-116" href="#FN-116"><sup>116</sup></a> +The Tasmanians of Oyster Bay use the native word of +similar meaning, <i>puggana</i>, man,<a id="FNA-117" href="#FN-117"><sup>117</sup></a> for 5.</p> + +<p>Wherever the numeral 20 is expressed by the term +<i>man</i>, it may be expected that 40 will be 2 men, 60, 3 +men, etc. This form of numeration is usually, though +not always, carried as far as the system extends; and +it sometimes leads to curious terms, of which a single +illustration will suffice. The San Blas Indians, like +almost all the other Central and South American tribes, +count by digit numerals, and form their twenties as +follows:<a id="FNA-118" href="#FN-118"><sup>118</sup></a></p> + +<table class="scale"> +<tr><td>20.</td><td>tula guena</td><td>= man 1.</td></tr> +<tr><td>40.</td><td>tula pogua</td><td>= man 2.</td></tr> +<tr><td>100.</td><td>tula atala</td><td>= man 5.</td></tr> +<tr><td>120.</td><td>tula nergua</td><td>= man 6.</td></tr> +<tr><td>1000.</td><td>tula wala guena</td><td>= great 1 man.</td></tr> +</table> + +<p>The last expression may, perhaps, be translated “great +hundred,” though the literal meaning is the one given. +<a class="pgnum" id="page-80" title="- 80 -"></a>If 10, instead of 20, is expressed by the word “man,” +the multiples of 10 follow the law just given for multiples +of 20. This is sufficiently indicated by the +Kusaie scale; or equally well by the Api words for +100 and 200, which are<a id="FNA-119" href="#FN-119"><sup>119</sup></a></p> + +<blockquote> +<p><i>duulimo toromomo</i> = 10 times the whole man.</p> +<p><i>duulimo toromomo va juo</i> = 10 times the whole man taken 2 times.</p> +</blockquote> + +<p>As an illustration of the legitimate result which is produced +by the attempt to express high numbers in this +manner the term applied by educated native Greenlanders<a id="FNA-120" href="#FN-120"><sup>120</sup></a> +for a thousand may be cited. This numeral, +which is, of course, not in common use, is</p> + +<blockquote> +<p><i>inuit kulit tatdlima nik kuleriartut navdlugit</i> = 10 men 5 times 10 times +come to an end.</p> +</blockquote> + +<p>It is worth noting that the word “great,” which appears +in the scale of the San Blas Indians, is not infrequently +made use of in the formation of higher numeral +words. The African Mabas<a id="FNA-121" href="#FN-121"><sup>121</sup></a> call 10 <i>atuk</i>, great 1; the +Hottentots<a id="FNA-122" href="#FN-122"><sup>122</sup></a> and the Hidatsa Indians call 100 great 10, +their words being <i>gei disi</i> and <i>pitikitstia</i> respectively.</p> + +<p>The Nicaraguans<a id="FNA-123" href="#FN-123"><sup>123</sup></a> express 100 by <i>guhamba</i>, great 10, +and 400 by <i>dinoamba</i>, great 20; and our own familiar +word “million,” which so many modern languages have +borrowed from the Italian, is nothing more nor less +<a class="pgnum" id="page-81" title="- 81 -"></a>than a derivative of the Latin <i>mille</i>, and really means +“great thousand.” The Dakota<a id="FNA-124" href="#FN-124"><sup>124</sup></a> language shows the +same origin for its expression of 1,000,000, which is <i>kick +ta opong wa tunkah</i>, great 1000. The origin of such +terms can hardly be ascribed to poverty of language. +It is found, rather, in the mental association of the +larger with the smaller unit, and the consequent repetition +of the name of the smaller. Any unit, whether +it be a single thing, a dozen, a score, a hundred, a +thousand, or any other unit, is, whenever used, a single +and complete group; and where the relation between +them is sufficiently close, as in our “gross” and “great +gross,” this form of nomenclature is natural enough to +render it a matter of some surprise that it has not +been employed more frequently. An old English +nursery rhyme makes use of this association, only in +a manner precisely the reverse of that which appears +now and then in numeral terms. In the latter case the +process is always one of enlargement, and the associative +word is “great.” In the following rhyme, constructed +by the mature for the amusement of the +childish mind, the process is one of diminution, and +the associative word is “little”:</p> + +<div class="poem"> +<p>One's none,</p> +<p>Two's some,</p> +<p>Three's a many,</p> +<p>Four's a penny,</p> +<p>Five's a little hundred.<a id="FNA-125" href="#FN-125"><sup>125</sup></a></p> +</div> + +<p class="noindent"><a class="pgnum" id="page-82" title="- 82 -"></a>Any real numeral formation by the use of “little,” +with the name of some higher unit, would, of course, +be impossible. The numeral scale must be complete +before the nursery rhyme can be manufactured.</p> + +<p>It is not to be supposed from the observations that +have been made on the formation of savage numeral +scales that all, or even the majority of tribes, proceed +in the awkward and faltering manner indicated by +many of the examples quoted. Some of the North +American Indian tribes have numeral scales which +are, as far as they go, as regular and almost as simple +as our own. But where digital numeration is extensively +resorted to, the expressions for higher numbers +are likely to become complex, and to act as a real bar +to the extension of the system. The same thing is +true, to an even greater degree, of tribes whose number +sense is so defective that they begin almost from +the outset to use combinations. If a savage expresses +the number 3 by the combination 2-1, it will at once +be suspected that his numerals will, by the time he +reaches 10 or 20, become so complex and confused that +numbers as high as these will be expressed by finger +pantomime rather than by words. Such is often the +case; and the comment is frequently made by explorers +that the tribes they have visited have no words for +numbers higher than 3, 4, 5, 10, or 20, but that counting +is carried beyond that point by the aid of fingers +or other objects. So reluctant, in many cases, are savages +to count by words, that limits have been assigned +<a class="pgnum" id="page-83" title="- 83 -"></a>for spoken numerals, which subsequent investigation +proved to fall far short of the real extent of the number +systems to which they belonged. One of the south-western +Indian tribes of the United States, the Comanches, +was for a time supposed to have no numeral +words below 10, but to count solely by the use of +fingers. But the entire scale of this taciturn tribe was +afterward discovered and published.</p> + +<p>To illustrate the awkward and inconvenient forms of +expression which abound in primitive numeral nomenclature, +one has only to draw from such scales as those +of the Zuñi, or the Point Barrow Eskimos, given in the +last chapter. Terms such as are found there may +readily be duplicated from almost any quarter of the +globe. The Soussous of Sierra Leone<a id="FNA-126" href="#FN-126"><sup>126</sup></a> call 99 <i>tongo +solo manani nun solo manani</i>, <i>i.e.</i> to take (10 understood) +5 + 4 times and 5 + 4. The Malagasy expression +for 1832 is<a id="FNA-127" href="#FN-127"><sup>127</sup></a> <i>roambistelo polo amby valonjato amby +arivo</i>, 2 + 30 + 800 + 1000. The Aztec equivalent for 399 +is<a id="FNA-128" href="#FN-128"><sup>128</sup></a> <i>caxtolli onnauh poalli ipan caxtolli onnaui</i>, (15 + 4) + × 20 + 15 + 4; and the Sioux require for 29 the ponderous +combination<a id="FNA-129" href="#FN-129"><sup>129</sup></a> <i>wick a chimen ne nompah sam pah +nep e chu wink a.</i> These terms, long and awkward as +they seem, are only the legitimate results which arise +from combining the names of the higher and lower +<a class="pgnum" id="page-84" title="- 84 -"></a>numbers, according to the peculiar genius of each language. +From some of the Australian tribes are derived +expressions still more complex, as for 6, <i>marh-jin-bang-ga-gudjir-gyn</i>, +half the hands and 1; and for 15, <i>marh-jin-belli-belli-gudjir-jina-bang-ga</i>, +the hand on either side and +half the feet.<a id="FNA-130" href="#FN-130"><sup>130</sup></a> The Maré tribe, one of the numerous +island tribes of Melanesia,<a id="FNA-131" href="#FN-131"><sup>131</sup></a> required for a translation +of the numeral 38, which occurs in John v. 5, “had an +infirmity thirty and eight years,” the circumlocution, +“one man and both sides five and three.” Such expressions, +curious as they seem at first thought, are no more +than the natural outgrowth of systems built up by the +slow and tedious process which so often obtains among +primitive races, where digit numerals are combined in an +almost endless variety of ways, and where mere reduplication +often serves in place of any independent names for +higher units. To what extent this may be carried is +shown by the language of the Cayubabi,<a id="FNA-132" href="#FN-132"><sup>132</sup></a> who have for +10 the word <i>tunca</i>, and for 100 and 1000 the compounds +<i>tunca tunca</i>, and <i>tunca tunca tunca</i> respectively; or of the +Sapibocones, who call 10 <i>bururuche</i>, hand hand, and 100 <i>buruche +buruche</i>, hand hand hand hand.<a id="FNA-133" href="#FN-133"><sup>133</sup></a> More remarkable +still is the Ojibwa language, which continues its numeral +scale without limit, furnishing combinations which are +really remarkable; as, <i>e.g.</i>, that for 1,000,000,000, which +is <i>me das wac me das wac as he me das wac</i>,<a id="FNA-134" href="#FN-134"><sup>134</sup></a> 1000 × 1000 +<a class="pgnum" id="page-85" title="- 85 -"></a> × 1000. The Winnebago expression for the same number,<a id="FNA-135" href="#FN-135"><sup>135</sup></a> +<i>ho ke he hhuta hhu chen a ho ke he ka ra pa ne za</i> +is no less formidable, but it has every appearance of +being an honest, native combination. All such primitive +terms for larger numbers must, however, be received +with caution. Savages are sometimes eager to display a +knowledge they do not possess, and have been known to +invent numeral words on the spot for the sake of carrying +their scales to as high a limit as possible. The +Choctaw words for million and billion are obvious attempts +to incorporate the corresponding English terms +into their own language.<a id="FNA-136" href="#FN-136"><sup>136</sup></a> For million they gave the +vocabulary-hunter the phrase <i>mil yan chuffa</i>, and for billion, +<i>bil yan chuffa</i>. The word <i>chuffa</i> signifies 1, hence +these expressions are seen at a glance to be coined solely +for the purpose of gratifying a little harmless Choctaw +vanity. But this is innocence itself compared with the +fraud perpetrated on Labillardière by the Tonga Islanders, +who supplied the astonished and delighted investigator +with a numeral vocabulary up to quadrillions. Their +real limit was afterward found to be 100,000, and above +that point they had palmed off as numerals a tolerably +complete list of the obscene words of their language, +together with a few nonsense terms. These were all +accepted and printed in good faith, and the humiliating +truth was not discovered until years afterward.<a id="FNA-137" href="#FN-137"><sup>137</sup></a></p> + +<p>One noteworthy and interesting fact relating to +<a class="pgnum" id="page-86" title="- 86 -"></a>numeral nomenclature is the variation in form which +words of this class undergo when applied to different +classes of objects. To one accustomed as we are to +absolute and unvarying forms for numerals, this seems +at first a novel and almost unaccountable linguistic +freak. But it is not uncommon among uncivilized +races, and is extensively employed by so highly enlightened +a people, even, as the Japanese. This variation +in form is in no way analogous to that produced by +inflectional changes, such as occur in Hebrew, Greek, +Latin, etc. It is sufficient in many cases to produce +almost an entire change in the form of the word; or +to result in compounds which require close scrutiny for +the detection of the original root. For example, in +the Carrier, one of the Déné dialects of western Canada, +the word <i>tha</i> means 3 things; <i>thane</i>, 3 persons; <i>that</i>, +3 times; <i>thatoen</i>, in 3 places; <i>thauh</i>, in 3 ways; <i>thailtoh</i>, +all of the 3 things; <i>thahoeltoh</i>, all of the 3 persons; +and <i>thahultoh</i>, all of the 3 times.<a id="FNA-138" href="#FN-138"><sup>138</sup></a> In the Tsimshian +language of British Columbia we find seven distinct +sets of numerals “which are used for various classes of +objects that are counted. The first set is used in +counting where there is no definite object referred to; +the second class is used for counting flat objects and +animals; the third for counting round objects and +divisions of time; the fourth for counting men; the +fifth for counting long objects, the numerals being composed +<a class="pgnum" id="page-87" title="- 87 -"></a>with <i>kan</i>, tree; the sixth for counting canoes; +and the seventh for measures. The last seem to be +composed with <i>anon</i>, hand.”<a id="FNA-139" href="#FN-139"><sup>139</sup></a> The first ten numerals +of each of these classes is given in the following table:</p> + +<table class="border"> +<thead> +<tr><th>No.</th><th>Counting</th><th>Flat<br/>Objects</th><th>Round<br/>Objects</th><th>Men</th><th>Long<br/>Objects</th><th>Canoes</th><th>Measures</th></tr> +</thead> +<tr><td class="number">1</td><td>gyak</td><td>gak</td><td>g'erel</td><td>k'al</td><td>k'awutskan</td><td>k'amaet</td><td>k'al</td></tr> +<tr><td class="number">2</td><td>t'epqat</td><td>t'epqat</td><td>goupel</td><td>t'epqadal</td><td>gaopskan</td><td>g'alpēeltk</td><td>gulbel</td></tr> +<tr><td class="number">3</td><td>guant</td><td>guant</td><td>gutle</td><td>gulal</td><td>galtskan</td><td>galtskantk</td><td>guleont</td></tr> +<tr><td class="number">4</td><td>tqalpq</td><td>tqalpq</td><td>tqalpq</td><td>tqalpqdal</td><td>tqaapskan</td><td>tqalpqsk</td><td>tqalpqalont</td></tr> +<tr><td class="number">5</td><td>kctōnc</td><td>kctōnc</td><td>kctōnc</td><td>kcenecal</td><td>k'etoentskan</td><td>kctōonsk</td><td>kctonsilont</td></tr> +<tr><td class="number">6</td><td>k'alt</td><td>k'alt</td><td>k'alt</td><td>k'aldal</td><td>k'aoltskan</td><td>k'altk</td><td>k'aldelont</td></tr> +<tr><td class="number">7</td><td>t'epqalt</td><td>t'epqalt</td><td>t'epqalt</td><td>t'epqaldal</td><td>t'epqaltskan</td><td>t'epqaltk</td><td>t'epqaldelont</td></tr> +<tr><td class="number">8</td><td>guandalt</td><td>yuktalt</td><td>yuktalt</td><td>yuktleadal</td><td>ek'tlaedskan</td><td>yuktaltk</td><td>yuktaldelont</td></tr> +<tr><td class="number">9</td><td>kctemac</td><td>kctemac</td><td>kctemac</td><td>kctemacal</td><td>kctemaestkan</td><td>kctemack</td><td>kctemasilont</td></tr> +<tr><td class="number">10</td><td>gy'ap</td><td>gy'ap</td><td>kpēel</td><td>kpal</td><td>kpēetskan</td><td>gy'apsk</td><td>kpeont</td></tr> +</table> + + +<p><a class="pgnum" id="page-88" title="- 88 -"></a>Remarkable as this list may appear, it is by no +means as extensive as that derived from many of the +other British Columbian tribes. The numerals of the +Shushwap, Stlatlumh, Okanaken, and other languages +of this region exist in several different forms, and can +also be modified by any of the innumerable suffixes of +these tongues.<a id="FNA-140" href="#FN-140"><sup>140</sup></a> To illustrate the almost illimitable +number of sets that may be formed, a table is given +of “a few classes, taken from the Heiltsuk dialect.<a id="FNA-141" href="#FN-141"><sup>141</sup></a> +It appears from these examples that the number of +classes is unlimited.”</p> + +<table class="border"> +<thead> +<tr><th></th><th>One.</th><th>Two.</th><th>Three.</th></tr> +</thead> + +<tr><th>Animate.</th><td>menok</td><td>maalok</td><td>yutuk</td></tr> +<tr><th>Round.</th><td>menskam</td><td>masem</td><td>yutqsem</td></tr> +<tr><th>Long.</th><td>ments'ak</td><td>mats'ak</td><td>yututs'ak</td></tr> +<tr><th>Flat.</th><td>menaqsa</td><td>matlqsa</td><td>yutqsa</td></tr> +<tr><th>Day.</th><td>op'enequls</td><td>matlp'enequls</td><td>yutqp'enequls</td></tr> +<tr><th>Fathom.</th><td>op'enkh</td><td>matlp'enkh</td><td>yutqp'enkh</td></tr> +<tr><th>Grouped together.</th><td>——</td><td>matloutl</td><td>yutoutl</td></tr> +<tr><th>Groups of objects.</th><td>nemtsmots'utl</td><td>matltsmots'utl</td><td>yutqtsmots'utl</td></tr> +<tr><th>Filled cup.</th><td>menqtlala</td><td>matl'aqtlala</td><td>yutqtlala</td></tr> +<tr><th>Empty cup.</th><td>menqtla</td><td>matl'aqtla</td><td>yutqtla</td></tr> +<tr><th>Full box.</th><td>menskamala</td><td>masemala</td><td>yutqsemala</td></tr> +<tr><th>Empty box.</th><td>menskam</td><td>masem</td><td>yutqsem</td></tr> +<tr><th>Loaded canoe.</th><td>mentsake</td><td>mats'ake</td><td>yututs'ake</td></tr> +<tr><th>Canoe with crew.</th><td>ments'akis</td><td>mats'akla</td><td>yututs'akla</td></tr> +<tr><th>Together on beach.</th><td>——</td><td>maalis</td><td>——</td></tr> +<tr><th>Together in house, etc.</th><td>——</td><td>maalitl</td><td>——</td></tr> +</table> + + +<p><a class="pgnum" id="page-89" title="- 89 -"></a>Variation in numeral forms such as is exhibited in +the above tables is not confined to any one quarter of +the globe; but it is more universal among the British +Columbian Indians than among any other race, and it +is a more characteristic linguistic peculiarity of this +than of any other region, either in the Old World or +in the New. It was to some extent employed by the +Aztecs,<a id="FNA-142" href="#FN-142"><sup>142</sup></a> and its use is current among the Japanese; in +whose language Crawfurd finds fourteen different classes +of numerals “without exhausting the list.”<a id="FNA-143" href="#FN-143"><sup>143</sup></a></p> + +<p>In examining the numerals of different languages it +will be found that the tens of any ordinary decimal +scale are formed in the same manner as in English. +Twenty is simply 2 times 10; 30 is 3 times 10, and +so on. The word “times” is, of course, not expressed, +any more than in English; but the expressions briefly +are, 2 tens, 3 tens, etc. But a singular exception to +this method is presented by the Hebrew, and other of +the Semitic languages. In Hebrew the word for 20 +is the plural of the word for 10; and 30, 40, 50, etc. +to 90 are plurals of 3, 4, 5, 6, 7, 8, 9. These numerals +are as follows:<a id="FNA-144" href="#FN-144"><sup>144</sup></a></p> + +<table class="scale"> +<tr><td>10,</td><td>eser,</td><td class="number">20,</td><td>eserim,</td></tr> +<tr><td>3,</td><td>shalosh,</td><td class="number">30,</td><td>shaloshim,</td></tr> +<tr><td>4,</td><td>arba,</td><td class="number">40,</td><td>arbaim,</td></tr> +<tr><td>5,</td><td>chamesh,</td><td class="number">50,</td><td>chamishshim,</td></tr> +<tr><td>6,</td><td>shesh,</td><td class="number">60,</td><td>sheshshim,</td></tr> +<tr><td><a class="pgnum" id="page-90" title="- 90 -"></a>7,</td><td>sheba,</td><td class="number">70,</td><td>shibim,</td></tr> +<tr><td>8,</td><td>shemoneh<ins title="inserted missing comma">,</ins></td><td class="number">80,</td><td>shemonim,</td></tr> +<tr><td>9,</td><td>tesha,</td><td class="number">90,</td><td>tishim.</td></tr> +</table> + + +<p>The same formation appears in the numerals of the +ancient Phœnicians,<a id="FNA-145" href="#FN-145"><sup>145</sup></a> and seems, indeed, to be a well-marked +characteristic of the various branches of this +division of the Caucasian race. An analogous method +appears in the formation of the tens in the Bisayan,<a id="FNA-146" href="#FN-146"><sup>146</sup></a> +one of the Malay numeral scales, where 30, 40, … +90, are constructed from 3, 4, … 9, by adding the +termination <i>-an</i>.</p> + +<p>No more interesting contribution has ever been made +to the literature of numeral nomenclature than that in +which Dr. Trumbull embodies the results of his scholarly +research among the languages of the native Indian +tribes of this country.<a id="FNA-147" href="#FN-147"><sup>147</sup></a> As might be expected, we are +everywhere confronted with a digital origin, direct or +indirect, in the great body of the words examined. +But it is clearly shown that such a derivation cannot +be established for all numerals; and evidence collected +by the most recent research fully substantiates the position +taken by Dr. Trumbull. Nearly all the derivations +established are such as to remind us of the meanings +we have already seen recurring in one form or another +in language after language. Five is the end of the +finger count on one hand—as, the Micmac <i>nan</i>, and +<a class="pgnum" id="page-91" title="- 91 -"></a>Mohegan <i>nunon</i>, gone, or spent; the Pawnee <i>sihuks</i>, +hands half; the Dakota <i>zaptan</i>, hand turned down; +and the Massachusetts <i>napanna</i>, on one side. Ten is +the end of the finger count, but is not always expressed +by the “both hands” formula so commonly met with. +The Cree term for this number is <i>mitatat</i>, no further; +and the corresponding word in Delaware is <i>m'tellen</i>, no +more. The Dakota 10 is, like its 5, a straightening +out of the fingers which have been turned over in +counting, or <i>wickchemna</i>, spread out unbent. The same +is true of the Hidatsa <i>pitika</i>, which signifies a smoothing +out, or straightening. The Pawnee 4, <i>skitiks</i>, is +unusual, signifying as it does “all the fingers,” or more +properly, “the fingers of the hand.” The same meaning +attaches to this numeral in a few other languages +also, and reminds one of the habit some people have +of beginning to count on the forefinger and proceeding +from there to the little finger. Can this have been the +habit of the tribes in question? A suggestion of the +same nature is made by the Illinois and Miami words +for 8, <i>parare</i> and <i>polane</i>, which signify “nearly ended.” +Six is almost always digital in origin, though the derivation +may be indirect, as in the Illinois <i>kakatchui</i>, +passing beyond the middle; and the Dakota <i>shakpe</i>, +1 in addition. Some of these significations are well +matched by numerals from the Ewe scales of western +Africa, where we find the following:<a id="FNA-148" href="#FN-148"><sup>148</sup></a></p> + +<table class="scale"> +<tr><td><a class="pgnum" id="page-92" title="- 92 -"></a>1.</td><td>de</td><td>= a going, <i>i.e.</i> a beginning. (Cf. the Zuñi <i>töpinte</i>, taken to +start with.)</td></tr> +<tr><td>3.</td><td>eto</td><td>= the father (from the middle, or longest finger).</td></tr> +<tr><td>6.</td><td>ade</td><td>= the other going.</td></tr> +<tr><td>9.</td><td>asieke</td><td>= parting with the hands.</td></tr> +<tr><td>10.</td><td>ewo</td><td>= done.</td></tr> +</table> + +<p>In studying the names for 2 we are at once led away +from a strictly digital origin for the terms by which +this number is expressed. These names seem to come +from four different sources: (1) roots denoting separation +or distinction; (2) likeness, equality, or opposition; +(3) addition, <i>i.e.</i> putting to, or putting with; (4) coupling, +pairing, or matching. They are often related to, +and perhaps derived from, names of natural pairs, as +feet, hands, eyes, arms, or wings. In the Dakota and +Algonkin dialects 2 is almost always related to “arms” +or “hands,” and in the Athapaskan to “feet.” But the +relationship is that of common origin, rather than of +derivation from these pair-names. In the Puri and +Hottentot languages, 2 and “hand” are closely allied; +while in Sanskrit, 2 may be expressed by any one of +the words <i>kara</i>, hand, <i>bahu</i>, arm, <i>paksha</i>, wing, or <i>netra,</i> +eye.<a id="FNA-149" href="#FN-149"><sup>149</sup></a> Still more remote from anything digital in their +derivation are the following, taken at random from a +very great number of examples that might be cited to +illustrate this point. The Assiniboines call 7, <i>shak ko +we</i>, or <i>u she nah</i>, the odd number.<a id="FNA-150" href="#FN-150"><sup>150</sup></a> The Crow 1, <i>hamat,</i> +signifies “the least”;<a id="FNA-151" href="#FN-151"><sup>151</sup></a> the Mississaga 1, <i>pecik</i>, a very +<a class="pgnum" id="page-93" title="- 93 -"></a>small thing.<a id="FNA-152" href="#FN-152"><sup>152</sup></a> In Javanese, Malay, and Manadu, the +words for 1, which are respectively <i>siji</i>, <i>satu</i>, and +<i>sabuah</i>, signify 1 seed, 1 pebble, and 1 fruit respectively<a id="FNA-153" href="#FN-153"><sup>153</sup></a>—words +as natural and as much to be expected at the +beginning of a number scale as any finger name could +possibly be. Among almost all savage races one form +or another of palpable arithmetic is found, such as +counting by seeds, pebbles, shells, notches, or knots; +and the derivation of number words from these sources +can constitute no ground for surprise. The Marquesan +word for 4 is <i>pona</i>, knot, from the practice of tying +breadfruit in knots of 4. The Maori 10 is <i>tekau</i>, +bunch, or parcel, from the counting of yams and fish +by parcels of 10.<a id="FNA-154" href="#FN-154"><sup>154</sup></a> The Javanese call 25, <i>lawe</i>, a thread, +or string; 50, <i>ekat</i>, a skein of thread; 400, <i>samas</i>, a bit +of gold; 800, <i>domas</i>, 2 bits of gold.<a id="FNA-155" href="#FN-155"><sup>155</sup></a> The Macassar +and Butong term for 100 is <i>bilangan</i>, 1 tale or reckoning.<a id="FNA-156" href="#FN-156"><sup>156</sup></a> +The Aztec 20 is <i>cem pohualli</i>, 1 count; 400 is +<i>centzontli</i>, 1 hair of the head; and 8000 is <i>xiquipilli</i>, +sack.<a id="FNA-157" href="#FN-157"><sup>157</sup></a> This sack was of such a size as to contain 8000 +cacao nibs, or grains, hence the derivation of the word +in its numeral sense is perfectly natural. In Japanese +we find a large number of terms which, as applied to +the different units of the number scale, seem almost +<a class="pgnum" id="page-94" title="- 94 -"></a>purely fanciful. These words, with their meanings as +given by a Japanese lexicon, are as follows:</p> + +<table class="scale"> +<tr><td>10,000, or 10<sup class="power">4</sup>,</td><td>män</td><td>= enormous number.</td></tr> +<tr><td>10<sup class="power">8</sup>,</td><td>oku</td><td>= a compound of the words “man” and “mind.”</td></tr> +<tr><td>10<sup class="power">12</sup>,</td><td>chio</td><td>= indication, or symptom.</td></tr> +<tr><td>10<sup class="power">16</sup>,</td><td>kei</td><td>= capital city.</td></tr> +<tr><td>10<sup class="power">20</sup>,</td><td>si</td><td>= a term referring to grains.</td></tr> +<tr><td>10<sup class="power">24</sup>,</td><td>owi</td><td>= ——</td></tr> +<tr><td>10<sup class="power">28</sup>,</td><td>jio</td><td>= extent of land.</td></tr> +<tr><td>10<sup class="power">32</sup>,</td><td>ko</td><td>= canal.</td></tr> +<tr><td>10<sup class="power">36</sup>,</td><td>kan</td><td>= some kind of a body of water.</td></tr> +<tr><td>10<sup class="power">40</sup>,</td><td>sai</td><td>= justice.</td></tr> +<tr><td>10<sup class="power">44</sup>,</td><td>sā</td><td>= support.</td></tr> +<tr><td>10<sup class="power">48</sup>,</td><td>kioku</td><td>= limit, or more strictly, ultimate.</td></tr> +<tr><td>.01<sup class="power">2</sup>,</td><td>rin</td><td>= ——</td></tr> +<tr><td>.01<sup class="power">3</sup>,</td><td>mo</td><td>= hair (of some animal).</td></tr> +<tr><td>.01<sup class="power">4</sup>,</td><td>shi</td><td>= thread.</td></tr> +</table> + +<p>In addition to these, some of the lower fractional +values are described by words meaning “very small,” +“very fine thread,” “sand grain,” “dust,” and “very +vague.” Taken altogether, the Japanese number system +is the most remarkable I have ever examined, in the +extent and variety of the higher numerals with well-defined +descriptive names. Most of the terms employed +are such as to defy any attempt to trace the process +of reasoning which led to their adoption. It is not +improbable that the choice was, in some of these cases +at least, either accidental or arbitrary; but still, the +changes in word meanings which occur with the lapse +of time may have differentiated significations originally +alike, until no trace of kinship would appear to the +<a class="pgnum" id="page-95" title="- 95 -"></a>casual observer. Our numerals “score” and “gross” +are never thought of as having any original relation to +what is conveyed by the other meanings which attach +to these words. But the origin of each, which is easily +traced, shows that, in the beginning, there existed a +well-defined reason for the selection of these, rather +than other terms, for the numbers they now describe. +Possibly these remarkable Japanese terms may be +accounted for in the same way, though the supposition +is, for some reasons, quite improbable. The same may +be said for the Malagasy 1000, <i>alina</i>, which also means +“night,” and the Hebrew 6, <i>shesh</i>, which has the +additional signification “white marble,” and the stray +exceptions which now and then come to the light in +this or that language. Such terms as these may admit +of some logical explanation, but for the great mass of +numerals whose primitive meanings can be traced at +all, no explanation whatever is needed; the words are +self-explanatory, as the examples already cited show.</p> + +<p>A few additional examples of natural derivation may +still further emphasize the point just discussed. In +Bambarese the word for 10, <i>tank</i>, is derived directly +from <i>adang</i>, to count.<a id="FNA-158" href="#FN-158"><sup>158</sup></a> In the language of Mota, one of +the islands of Melanesia, 100 is <i>mel nol</i>, used and done +with, referring to the leaves of the cycas tree, with +which the count had been carried on.<a id="FNA-159" href="#FN-159"><sup>159</sup></a> In many other +Melanesian dialects<a id="FNA-160" href="#FN-160"><sup>160</sup></a> 100 is <i>rau</i>, a branch or leaf. In +<a class="pgnum" id="page-96" title="- 96 -"></a>the Torres Straits we find the same number expressed +by <i>na won</i>, the close; and in Eromanga it is <i>narolim +narolim</i> (2 × 5)(2 × 5).<a id="FNA-161" href="#FN-161"><sup>161</sup></a> This combination deserves +remark only because of the involved form which seems +to have been required for the expression of so small +a number as 100. A compound instead of a simple +term for any higher unit is never to be wondered at, +so rude are some of the savage methods of expressing +number; but “two fives (times) two fives” is certainly +remarkable. Some form like that employed by the Nusqually<a id="FNA-162" href="#FN-162"><sup>162</sup></a> +of Puget Sound for 1000, i.e. <i>paduts-subquätche</i>, +ten hundred, is more in accordance with primitive +method. But we are equally likely to find such descriptive +phrases for this numeral as the <i>dor paka</i>, banyan +roots, of the Torres Islands; <i>rau na hai</i>, leaves of a +tree, of Vaturana; or <i>udolu</i>, all, of the Fiji Islands. +And two curious phrases for 1000 are those of the +Banks' Islands, <i>tar mataqelaqela</i>, eye blind thousand, <i>i.e.</i> +many beyond count; and of Malanta, <i>warehune huto</i>, +opossum's hairs, or <i>idumie one</i>, count the sand.<a id="FNA-163" href="#FN-163"><sup>163</sup></a></p> + +<p>The native languages of India, Thibet, and portions +of the Indian archipelago furnish us with abundant +instances of the formation of secondary numeral scales, +which were used only for special purposes, and without +in any way interfering with the use of the number words +already in use. “Thus the scholars of India, ages ago, +<a class="pgnum" id="page-97" title="- 97 -"></a>selected a set of words for a memoria technica, in order +to record dates and numbers. These words they chose +for reasons which are still in great measure evident; +thus ‘moon’ or ‘earth’ expressed 1, there being but +one of each; 2 might be called ‘eye,’ ‘wing,’ ‘arm,’ +‘jaw,’ as going in pairs; for 3 they said ‘Rama,’ ‘fire,’ +or ‘quality,’ there being considered to be three Ramas, +three kinds of fire, three qualities (guna); for 4 were +used ‘veda,’ ‘age,’ or ‘ocean,’ there being four of each +recognized; ‘season’ for 6, because they reckoned six +seasons; ‘sage’ or ‘vowel,’ for 7, from the seven sages +and the seven vowels; and so on with higher numbers, +‘sun’ for 12, because of his twelve annual denominations, +or ‘zodiac’ from his twelve signs, and ‘nail’ for +20, a word incidentally bringing in finger notation. As +Sanskrit is very rich in synonyms, and as even the +numerals themselves might be used, it became very +easy to draw up phrases or nonsense verses to record +series of numbers by this system of artificial memory.”<a id="FNA-164" href="#FN-164"><sup>164</sup></a></p> + +<p>More than enough has been said to show how baseless +is the claim that all numeral words are derived, either +directly or indirectly, from the names of fingers, hands, +or feet. Connected with the origin of each number +word there may be some metaphor, which cannot always +be distinctly traced; and where the metaphor was born +of the hand or of the foot, we inevitably associate it +with the practice of finger counting. But races as fond +of metaphor and of linguistic embellishment as are those +<a class="pgnum" id="page-98" title="- 98 -"></a>of the East, or as are our American Indians even, might +readily resort to some other source than that furnished +by the members of the human body, when in want of +a term with which to describe the 5, 10, or any other +number of the numeral scale they were unconsciously +forming. That the first numbers of a numeral scale +are usually derived from other sources, we have some +reason to believe; but that all above 2, 3, or at most +4, are almost universally of digital origin we must admit. +Exception should properly be made of higher units, say +1000 or anything greater, which could not be expected +to conform to any law of derivation governing the first +few units of a system.</p> + +<p>Collecting together and comparing with one another +the great mass of terms by which we find any number +expressed in different languages, and, while admitting +the great diversity of method practised by different +tribes, we observe certain resemblances which were not +at first supposed to exist. The various meanings of 1, +where they can be traced at all, cluster into a little +group of significations with which at last we come to +associate the idea of unity. Similarly of 2, or 5, or 10, +or any one of the little band which does picket duty +for the advance guard of the great host of number +words which are to follow. A careful examination of +the first decade warrants the assertion that the probable +meaning of any one of the units will be found in +the list given below. The words selected are intended +merely to serve as indications of the thought underlying +<a class="pgnum" id="page-99" title="- 99 -"></a>the savage's choice, and not necessarily as the +exact term by means of which he describes his number. +Only the commonest meanings are included in +the tabulation here given.</p> + +<table class="scale"> +<tr><td>1</td><td>= existence, piece, group, beginning.</td></tr> +<tr><td>2</td><td>= repetition, division, natural pair.</td></tr> +<tr><td>3</td><td>= collection, many, two-one.</td></tr> +<tr><td>4</td><td>= two twos.</td></tr> +<tr><td>5</td><td>= hand, group, division,</td></tr> +<tr><td>6</td><td>= five-one, two threes, second one.</td></tr> +<tr><td>7</td><td>= five-two, second two, three from ten.</td></tr> +<tr><td>8</td><td>= five-three, second three, two fours, two from ten.</td></tr> +<tr><td>9</td><td>= five-four, three threes, one from ten.</td></tr> +<tr><td>10</td><td>= one (group), two fives (hands), half a man, one man.</td></tr> +<tr><td>15</td><td>= ten-five, one foot, three fives.</td></tr> +<tr><td>20</td><td>= two tens, one man, two feet.<a id="FNA-165" href="#FN-165"><sup>165</sup></a></td></tr> +</table> + + + + + +<h2><a class="pgnum" id="page-100" title="- 100 -"></a>Chapter V.</h2> +<h3>Miscellaneous Number Bases.</h3> + + +<p><span class="initial">In</span> the development and extension of any series of +numbers into a systematic arrangement to which the +term <i>system</i> may be applied, the first and most indispensable +step is the selection of some number which is +to serve as a base. When the savage begins the process +of counting he invents, one after another, names +with which to designate the successive steps of his +numerical journey. At first there is no attempt at +definiteness in the description he gives of any considerable +number. If he cannot show what he means by +the use of his fingers, or perhaps by the fingers of a +single hand, he unhesitatingly passes it by, calling it +many, heap, innumerable, as many as the leaves on the +trees, or something else equally expressive and equally +indefinite. But the time comes at last when a greater +degree of exactness is required. Perhaps the number +11 is to be indicated, and indicated precisely. A fresh +mental effort is required of the ignorant child of +nature; and the result is “all the fingers and one +more,” “both hands and one more,” “one on another +count,” or some equivalent circumlocution. If he has +an independent word for 10, the result will be simply +<a class="pgnum" id="page-101" title="- 101 -"></a>ten-one. When this step has been taken, the base is +established. The savage has, with entire unconsciousness, +made all his subsequent progress dependent on +the number 10, or, in other words, he has established +10 as the base of his number system. The process just +indicated may be gone through with at 5, or at 20, +thus giving us a quinary or a vigesimal, or, more probably, +a mixed system; and, in rare instances, some +other number may serve as the point of departure +from simple into compound numeral terms. But the +general idea is always the same, and only the details +of formation are found to differ.</p> + +<p>Without the establishment of some base any <i>system</i> +of numbers is impossible. The savage has no means of +keeping track of his count unless he can at each step +refer himself to some well-defined milestone in his +course. If, as has been pointed out in the foregoing +chapters, confusion results whenever an attempt is made +to count any number which carries him above 10, it +must at once appear that progress beyond that point +would be rendered many times more difficult if it were +not for the fact that, at each new step, he has only to +indicate the distance he has progressed beyond his base, +and not the distance from his original starting-point. +Some idea may, perhaps, be gained of the nature of +this difficulty by imagining the numbers of our ordinary +scale to be represented, each one by a single +symbol different from that used to denote any other +number. How long would it take the average intellect +<a class="pgnum" id="page-102" title="- 102 -"></a>to master the first 50 even, so that each number could +without hesitation be indicated by its appropriate symbol? +After the first 50 were once mastered, what of +the next 50? and the next? and the next? and so on. +The acquisition of a scale for which we had no other +means of expression than that just described would be +a matter of the extremest difficulty, and could never, +save in the most exceptional circumstances, progress +beyond the attainment of a limit of a few hundred. +If the various numbers in question were designated by +words instead of by symbols, the difficulty of the task +would be still further increased. Hence, the establishment +of some number as a base is not only a matter +of the very highest convenience, but of absolute necessity, +if any save the first few numbers are ever to +be used.</p> + +<p>In the selection of a base,—of a number from which +he makes a fresh start, and to which he refers the +next steps in his count,—the savage simply follows +nature when he chooses 10, or perhaps 5 or 20. But +it is a matter of the greatest interest to find that other +numbers have, in exceptional cases, been used for this +purpose. Two centuries ago the distinguished philosopher +and mathematician, Leibnitz, proposed a binary +system of numeration. The only symbols needed in +such a system would be 0 and 1. The number which +is now symbolized by the figure 2 would be represented +by 10; while 3, 4, 5, 6, 7, 8, etc., would appear +in the binary notation as 11, 100, 101, 110, 111, 1000, +<a class="pgnum" id="page-103" title="- 103 -"></a>etc. The difficulty with such a system is that it rapidly +grows cumbersome, requiring the use of so many +figures for indicating any number. But Leibnitz found +in the representation of all numbers by means of the +two digits 0 and 1 a fitting symbolization of the creation +out of chaos, or nothing, of the entire universe by +the power of the Deity. In commemoration of this +invention a medal was struck bearing on the obverse +the words</p> + +<blockquote> +<p class="ctr">Numero Deus impari gaudet,</p> +</blockquote> + +<p class="noindent">and on the reverse,</p> + +<blockquote> +<p class="ctr">Omnibus ex nihilo ducendis sufficit Unum.<a id="FNA-166" href="#FN-166"><sup>166</sup></a></p> +</blockquote> + +<p class="noindent">This curious system seems to have been regarded with +the greatest affection by its inventor, who used every +endeavour in his power to bring it to the notice of +scholars and to urge its claims. But it appears to have +been received with entire indifference, and to have +been regarded merely as a mathematical curiosity.</p> + +<p>Unknown to Leibnitz, however, a binary method of +counting actually existed during that age; and it is +only at the present time that it is becoming extinct. +In Australia, the continent that is unique in its flora, +its fauna, and its general topography, we find also this +anomaly among methods of counting. The natives, +who are to be classed among the lowest and the least +intelligent of the aboriginal races of the world, have +number systems of the most rudimentary nature, and +<a class="pgnum" id="page-104" title="- 104 -"></a>evince a decided tendency to count by twos. This +peculiarity, which was to some extent shared by the +Tasmanians, the island tribes of the Torres Straits, +and other aboriginal races of that region, has by some +writers been regarded as peculiar to their part of the +world; as though a binary number system were not +to be found elsewhere. This attempt to make out of +the rude and unusual method of counting which obtained +among the Australians a racial characteristic is +hardly justified by fuller investigation. Binary number +systems, which are given in full on another page, +are found in South America. Some of the Dravidian +scales are binary;<a id="FNA-167" href="#FN-167"><sup>167</sup></a> and the marked preference, not +infrequently observed among savage races, for counting +by pairs, is in itself a sufficient refutation of this +theory. Still it is an unquestionable fact that this +binary tendency is more pronounced among the Australians +than among any other extensive number of +kindred races. They seldom count in words above +4, and almost never as high as 7. One of the most +careful observers among them expresses his doubt as +to a native's ability to discover the loss of two pins, +if he were first shown seven pins in a row, and then +two were removed without his knowledge.<a id="FNA-168" href="#FN-168"><sup>168</sup></a> But he +believes that if a single pin were removed from the +seven, the Blackfellow would become conscious of its +loss. This is due to his habit of counting by pairs, +<a class="pgnum" id="page-105" title="- 105 -"></a>which enables him to discover whether any number +within reasonable limit is odd or even. Some of the +negro tribes of Africa, and of the Indian tribes of +America, have the same habit. Progression by pairs +may seem to some tribes as natural as progression by +single units. It certainly is not at all rare; and in +Australia its influence on spoken number systems is +most apparent.</p> + +<p>Any number system which passes the limit 10 is +reasonably sure to have either a quinary, a decimal, or +a vigesimal structure. A binary scale could, as it is +developed in primitive languages, hardly extend to 20, +or even to 10, without becoming exceedingly cumbersome. +A binary scale inevitably suggests a wretchedly +low degree of mental development, which stands in the +way of the formation of any number scale worthy to be +dignified by the name of system. Take, for example, +one of the dialects found among the western tribes of +the Torres Straits, where, in general, but two numerals +are found to exist. In this dialect the method of counting +is:<a id="FNA-169" href="#FN-169"><sup>169</sup></a></p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">urapun.</td></tr> +<tr><td>2.</td><td colspan="2">okosa.</td></tr> +<tr><td>3.</td><td>okosa urapun</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>okosa okosa</td><td>= 2-2.</td></tr> +<tr><td>5.</td><td>okosa okosa urapun</td><td>= 2-2-1.</td></tr> +<tr><td>6.</td><td>okosa okosa okosa</td><td>= 2-2-2.</td></tr> +</table> + +<p class="noindent">Anything above 6 they call <i>ras</i>, a lot.</p> + +<p>For the sake of uniformity we may speak of this as +a “system.” But in so doing, we give to the legitimate +<a class="pgnum" id="page-106" title="- 106 -"></a>meaning of the word a severe strain. The customs and +modes of life of these people are not such as to require +the use of any save the scanty list of numbers given +above; and their mental poverty prompts them to call 3, +the first number above a single pair, 2-1. In the same +way, 4 and 6 are respectively 2 pairs and 3 pairs, while +5 is 1 more than 2 pairs. Five objects, however, they +sometimes denote by <i>urapuni-getal</i>, 1 hand. A precisely +similar condition is found to prevail respecting the arithmetic +of all the Australian tribes. In some cases only +two numerals are found, and in others three. But in +a very great number of the native languages of that +continent the count proceeds by pairs, if indeed it proceeds +at all. Hence we at once reject the theory that +Australian arithmetic, or Australian counting, is essentially +peculiar. It is simply a legitimate result, such +as might be looked for in any part of the world, of the +barbarism in which the races of that quarter of the world +were sunk, and in which they were content to live.</p> + +<p>The following examples of Australian and Tasmanian +number systems show how scanty was the numerical +ability possessed by these tribes, and illustrate fully +their tendency to count by twos or pairs.</p> + +<table class="scale"> +<caption>Murray River.<a id="FNA-170" href="#FN-170"><sup>170</sup></a></caption> +<tr><td>1.</td><td colspan="2">enea.</td></tr> +<tr><td>2.</td><td colspan="2">petcheval.</td></tr> +<tr><td>3.</td><td>petchevalenea</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>petcheval peteheval</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Maroura.</caption> +<tr><td>1.</td><td colspan="2">nukee.</td></tr> +<tr><td>2.</td><td colspan="2">barkolo.</td></tr> +<tr><td>3.</td><td>barkolo nuke</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>barkolo barkolo</td><td>= 2-2.</td></tr> +</table> + + +<table class="scale"> +<caption><a class="pgnum" id="page-107" title="- 107 -"></a>Lake Kopperamana.</caption> +<tr><td>1.</td><td colspan="2">ngerna.</td></tr> +<tr><td>2.</td><td colspan="2">mondroo.</td></tr> +<tr><td>3.</td><td colspan="2">barkooloo.</td></tr> +<tr><td>4.</td><td>mondroo mondroo</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Mort Noular.</caption> +<tr><td>1.</td><td colspan="2">gamboden.</td></tr> +<tr><td>2.</td><td colspan="2">bengeroo.</td></tr> +<tr><td>3.</td><td>bengeroganmel</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>bengeroovor bengeroo</td><td>= 2 + 2.</td></tr> +</table> + +<table class="scale"> +<caption>Wimmera.</caption> +<tr><td>1.</td><td colspan="2">keyap.</td></tr> +<tr><td>2.</td><td colspan="2">pollit.</td></tr> +<tr><td>3.</td><td>pollit keyap</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>pollit pollit</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Popham Bay.</caption> +<tr><td>1.</td><td colspan="2">motu.</td></tr> +<tr><td>2.</td><td colspan="2">lawitbari.</td></tr> +<tr><td>3.</td><td>lawitbari-motu</td><td>= 2-1.</td></tr> +</table> + +<table class="scale"> +<caption>Kamilaroi.<a id="FNA-171" href="#FN-171"><sup>171</sup></a></caption> +<tr><td>1.</td><td colspan="2">mal.</td></tr> +<tr><td>2.</td><td colspan="2">bularr.</td></tr> +<tr><td>3.</td><td colspan="2">guliba.</td></tr> +<tr><td>4.</td><td>bularrbularr</td><td>= 2-2.</td></tr> +<tr><td>5.</td><td>bulaguliba</td><td>= 2-3.</td></tr> +<tr><td>6.</td><td>gulibaguliba</td><td>= 3-3.</td></tr> +</table> + +<table class="scale"> +<caption>Port Essington.<a id="FNA-172" href="#FN-172"><sup>172</sup></a></caption> +<tr><td>1.</td><td colspan="2">erad.</td></tr> +<tr><td>2.</td><td colspan="2">nargarik.</td></tr> +<tr><td>3.</td><td>nargarikelerad</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>nargariknargarik</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Warrego.</caption> +<tr><td>1.</td><td colspan="2">tarlina.</td></tr> +<tr><td>2.</td><td colspan="2">barkalo.</td></tr> +<tr><td>3.</td><td>tarlina barkalo</td><td>= 1-2.</td></tr> +</table> + +<table class="scale"> +<caption>Crocker Island.</caption> +<tr><td>1.</td><td colspan="2">roka.</td></tr> +<tr><td>2.</td><td colspan="2">orialk.</td></tr> +<tr><td>3.</td><td>orialkeraroka</td><td>= 2-1.</td></tr> +</table> + +<table class="scale"> +<caption>Warrior Island.<a id="FNA-173" href="#FN-173"><sup>173</sup></a></caption> +<tr><td>1.</td><td colspan="2">woorapoo.</td></tr> +<tr><td>2.</td><td colspan="2">ocasara.</td></tr> +<tr><td>3.</td><td>ocasara woorapoo</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>ocasara ocasara</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Dippil.<a id="FNA-174" href="#FN-174"><sup>174</sup></a></caption> +<tr><td>1.</td><td colspan="2">kalim.</td></tr> +<tr><td>2.</td><td colspan="2">buller.</td></tr> +<tr><td>3.</td><td colspan="2">boppa.</td></tr> +<tr><td>4.</td><td>buller gira buller</td><td>= 2 + 2.</td></tr> +<tr><td>5.</td><td>buller gira buller kalim</td><td>= 2 + 2 + 1.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-108" title="- 108 -"></a>Frazer's Island.<a id="FNA-175" href="#FN-175"><sup>175</sup></a></caption> +<tr><td>1.</td><td colspan="2">kalim.</td></tr> +<tr><td>2.</td><td colspan="2">bulla.</td></tr> +<tr><td>3.</td><td colspan="2">goorbunda.</td></tr> +<tr><td>4.</td><td>bulla-bulla</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Moreton's Bay.<a id="FNA-176" href="#FN-176"><sup>176</sup></a></caption> + +<tr><td>1.</td><td colspan="2">kunner.</td></tr> +<tr><td>2.</td><td colspan="2">budela.</td></tr> +<tr><td>3.</td><td colspan="2">muddan.</td></tr> +<tr><td>4.</td><td>budela berdelu</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Encounter Bay.<a id="FNA-177" href="#FN-177"><sup>177</sup></a></caption> +<tr><td>1.</td><td colspan="2">yamalaitye.</td></tr> +<tr><td>2.</td><td colspan="2">ningenk.</td></tr> +<tr><td>3.</td><td colspan="2">nepaldar.</td></tr> +<tr><td>4.</td><td>kuko kuko</td><td>= 2-2, or pair pair.</td></tr> +<tr><td>5.</td><td>kuko kuko ki</td><td>= 2-2-1.</td></tr> +<tr><td>6.</td><td>kuko kuko kuko</td><td>= 2-2-2.</td></tr> +<tr><td>7.</td><td>kuko kuko kuko ki</td><td>= 2-2-2-1.</td></tr> +</table> + +<table class="scale"> +<caption>Adelaide.<a id="FNA-178" href="#FN-178"><sup>178</sup></a></caption> + +<tr><td>1.</td><td colspan="2">kuma.</td></tr> +<tr><td>2.</td><td colspan="2">purlaitye, or bula.</td></tr> +<tr><td>3.</td><td colspan="2">marnkutye.</td></tr> +<tr><td>4.</td><td>yera-bula</td><td>= pair 2.</td></tr> +<tr><td>5.</td><td>yera-bula kuma</td><td>= pair 2-1.</td></tr> +<tr><td>6.</td><td>yera-bula purlaitye</td><td>= pair 2.2.</td></tr> +</table> + +<table class="scale"> +<caption>Wiraduroi.<a id="FNA-179" href="#FN-179"><sup>179</sup></a></caption> + +<tr><td>1.</td><td colspan="2">numbai.</td></tr> +<tr><td>2.</td><td colspan="2">bula.</td></tr> +<tr><td>3.</td><td>bula-numbai</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>bungu</td><td>= many.</td></tr> +<tr><td>5.</td><td>bungu-galan</td><td>= very many.</td></tr> +</table> + +<table class="scale"> +<caption>Wirri-Wirri.<a id="FNA-180" href="#FN-180"><sup>180</sup></a></caption> + +<tr><td>1.</td><td colspan="2">mooray.</td></tr> +<tr><td>2.</td><td colspan="2">boollar.</td></tr> +<tr><td>3.</td><td>belar mooray</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>boollar boollar</td><td>= 2-2.</td></tr> +<tr><td>5.</td><td colspan="2">mongoonballa.</td></tr> +<tr><td>6.</td><td colspan="2">mongun mongun.</td></tr> +</table> + +<table class="scale"> +<caption>Cooper's Creek.<a id="FNA-181" href="#FN-181"><sup>181</sup></a></caption> + +<tr><td>1.</td><td colspan="2">goona.</td></tr> +<tr><td>2.</td><td colspan="2">barkoola.</td></tr> +<tr><td>3.</td><td>barkoola goona</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>barkoola barkoola</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Bourke, Darling River.<a id="FNA-182" href="#FN-182"><sup>182</sup></a></caption> + +<tr><td>1.</td><td colspan="2">neecha.</td></tr> +<tr><td>2.</td><td colspan="2">boolla.</td></tr> +<tr><td>4.</td><td>boolla neecha</td><td>= 2-1.</td></tr> +<tr><td>3.</td><td>boolla boolla</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-109" title="- 109 -"></a>Murray River, N.W. Bend.<a id="FNA-183" href="#FN-183"><sup>183</sup></a></caption> +<tr><td>1.</td><td colspan="2">mata.</td></tr> +<tr><td>2.</td><td colspan="2">rankool.</td></tr> +<tr><td>3.</td><td>rankool mata</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>rankool rankool</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Yit-tha.<a id="FNA-184" href="#FN-184"><sup>184</sup></a></caption> +<tr><td>1.</td><td colspan="2">mo.</td></tr> +<tr><td>2.</td><td colspan="2">thral.</td></tr> +<tr><td>3.</td><td>thral mo</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>thral thral</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Port Darwin.<a id="FNA-185" href="#FN-185"><sup>185</sup></a></caption> +<tr><td>1.</td><td colspan="2">kulagook.</td></tr> +<tr><td>2.</td><td colspan="2">kalletillick.</td></tr> +<tr><td>3.</td><td>kalletillick kulagook</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>kalletillick kalletillick</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Champion Bay.<a id="FNA-186" href="#FN-186"><sup>186</sup></a></caption> +<tr><td>1.</td><td colspan="2">kootea.</td></tr> +<tr><td>2.</td><td colspan="2">woothera.</td></tr> +<tr><td>3.</td><td>woothera kootea</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>woothera woothera</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Belyando River.<a id="FNA-187" href="#FN-187"><sup>187</sup></a></caption> +<tr><td>1.</td><td colspan="2">wogin.</td></tr> +<tr><td>2.</td><td colspan="2">booleroo.</td></tr> +<tr><td>3.</td><td>booleroo wogin</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>booleroo booleroo</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Warrego River.</caption> +<tr><td>1.</td><td colspan="2">onkera.</td></tr> +<tr><td>2.</td><td colspan="2">paulludy.</td></tr> +<tr><td>3.</td><td>paulludy onkera</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>paulludy paulludy</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Richmond River.</caption> +<tr><td>1.</td><td colspan="2">yabra.</td></tr> +<tr><td>2.</td><td colspan="2">booroora.</td></tr> +<tr><td>3.</td><td>booroora yabra</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>booroora booroora</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Port Macquarie.</caption> +<tr><td>1.</td><td colspan="2">warcol.</td></tr> +<tr><td>2.</td><td colspan="2">blarvo.</td></tr> +<tr><td>3.</td><td>blarvo warcol</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>blarvo blarvo</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Hill End.</caption> +<tr><td>1.</td><td colspan="2">miko.</td></tr> +<tr><td>2.</td><td colspan="2">bullagut.</td></tr> +<tr><td>3.</td><td>bullagut miko</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>bullagut bullagut</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Moneroo</caption> +<tr><td>1.</td><td colspan="2">boor.</td></tr> +<tr><td>2.</td><td colspan="2">wajala, blala.</td></tr> +<tr><td>3.</td><td>blala boor</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td colspan="2">wajala wajala.</td></tr> +</table> + + +<table class="scale"> +<caption><a class="pgnum" id="page-110" title="- 110 -"></a>Gonn Station.</caption> +<tr><td>1.</td><td colspan="2">karp.</td></tr> +<tr><td>2.</td><td colspan="2">pellige.</td></tr> +<tr><td>3.</td><td>pellige karp</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>pellige pellige</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Upper Yarra.</caption> +<tr><td>1.</td><td colspan="2">kaambo.</td></tr> +<tr><td>2.</td><td colspan="2">benjero.</td></tr> +<tr><td>3.</td><td>benjero kaambo</td><td>= 2-2.</td></tr> +<tr><td>4.</td><td>benjero on benjero</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Omeo.</caption> +<tr><td>1.</td><td colspan="2">bore.</td></tr> +<tr><td>2.</td><td colspan="2">warkolala.</td></tr> +<tr><td>3.</td><td>warkolala bore</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>warkolala warkolala</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Snowy River.</caption> +<tr><td>1.</td><td colspan="2">kootook.</td></tr> +<tr><td>2.</td><td colspan="2">boolong.</td></tr> +<tr><td>3.</td><td>booloom catha kootook</td><td>= 2 + 1.</td></tr> +<tr><td>4.</td><td>booloom catha booloom</td><td>= 2 + 2.</td></tr> +</table> + +<table class="scale"> +<caption>Ngarrimowro.</caption> +<tr><td>1.</td><td colspan="2">warrangen.</td></tr> +<tr><td>2.</td><td colspan="2">platir.</td></tr> +<tr><td>3.</td><td>platir warrangen</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>platir platir</td><td>= 2-2.</td></tr> +</table> + +<p>This Australian list might be greatly extended, but +the scales selected may be taken as representative +examples of Australian binary scales. Nearly all of +them show a structure too clearly marked to require +comment. In a few cases, however, the systems are +to be regarded rather as showing a trace of binary +structure, than as perfect examples of counting by +twos. Examples of this nature are especially numerous +in Curr's extensive list—the most complete collection +of Australian vocabularies ever made.</p> + +<p>A few binary scales have been found in South +America, but they show no important variation on the +Australian systems cited above. The only ones I have +been able to collect are the following:</p> + +<table class="scale"> +<caption><a class="pgnum" id="page-111" title="- 111 -"></a>Bakairi.<a id="FNA-188" href="#FN-188"><sup>188</sup></a></caption> +<tr><td>1.</td><td colspan="2">tokalole.</td></tr> +<tr><td>2.</td><td colspan="2">asage.</td></tr> +<tr><td>3.</td><td>asage tokalo</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>asage asage</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Zapara.<a id="FNA-189" href="#FN-189"><sup>189</sup></a></caption> +<tr><td>1.</td><td colspan="2">nuquaqui.</td></tr> +<tr><td>2.</td><td colspan="2">namisciniqui.</td></tr> +<tr><td>3.</td><td colspan="2">haimuckumarachi.</td></tr> +<tr><td>4.</td><td>namisciniqui ckara maitacka</td><td>= 2 + 2.</td></tr> +<tr><td>5.</td><td>namisciniqui ckara maitacka nuquaqui</td><td>= 2 pairs + 1.</td></tr> +<tr><td>6.</td><td>haimuckumaracki ckaramsitacka</td><td>= 3 pairs.</td></tr> +</table> + +<table class="scale"> +<caption>Apinages.<a id="FNA-190" href="#FN-190"><sup>190</sup></a></caption> +<tr><td>1.</td><td colspan="2">pouchi.</td></tr> +<tr><td>2.</td><td colspan="2">at croudou.</td></tr> +<tr><td>3.</td><td>at croudi-pshi</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>agontad-acroudo</td><td>= 2-2.</td></tr> +</table> + +<table class="scale"> +<caption>Cotoxo.<a id="FNA-191" href="#FN-191"><sup>191</sup></a></caption> +<tr><td>1.</td><td colspan="2">ihueto.</td></tr> +<tr><td>2.</td><td colspan="2">ize.</td></tr> +<tr><td>3.</td><td>ize-te-hueto</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>ize-te-seze</td><td>= 2-2.</td></tr> +<tr><td>5.</td><td>ize-te-seze-hue</td><td>= 2-2-1.</td></tr> +</table> + +<table class="scale"> +<caption>Mbayi.<a id="FNA-192" href="#FN-192"><sup>192</sup></a></caption> +<tr><td>1.</td><td colspan="2">uninitegui.</td></tr> +<tr><td>2.</td><td colspan="2">iniguata.</td></tr> +<tr><td>3.</td><td>iniguata dugani</td><td>= 2 over.</td></tr> +<tr><td>4.</td><td>iniguata driniguata</td><td>= 2-2.</td></tr> +<tr><td>5.</td><td>oguidi</td><td>= many.</td></tr> +</table> + +<table class="scale"> +<caption>Tama.<a id="FNA-193" href="#FN-193"><sup>193</sup></a></caption> +<tr><td>1.</td><td colspan="2">teyo.</td></tr> +<tr><td>2.</td><td colspan="2">cayapa.</td></tr> +<tr><td>3.</td><td>cho-teyo</td><td>= 2 + 1.</td></tr> +<tr><td>4.</td><td>cayapa-ria</td><td>= 2 again.</td></tr> +<tr><td>5.</td><td>cia-jente</td><td>= hand.</td></tr> +</table> + +<table class="scale"> +<caption>Curetu.<a id="FNA-194" href="#FN-194"><sup>194</sup></a></caption> +<tr><td>1.</td><td colspan="2">tchudyu.</td></tr> +<tr><td>2.</td><td colspan="2">ap-adyu.</td></tr> +<tr><td>3.</td><td colspan="2">arayu.</td></tr> +<tr><td>4.</td><td>apaedyái</td><td>= 2 + 2.</td></tr> +<tr><td>5.</td><td colspan="2">tchumupa.</td></tr> +</table> + + +<p>If the existence of number systems like the above are +to be accounted for simply on the ground of low civilization, +<a class="pgnum" id="page-112" title="- 112 -"></a>one might reasonably expect to find ternary and +and quaternary scales, as well as binary. Such scales +actually exist, though not in such numbers as the binary. +An example of the former is the Betoya scale,<a id="FNA-195" href="#FN-195"><sup>195</sup></a> which +runs thus:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">edoyoyoi.</td></tr> +<tr><td>2.</td><td>edoi</td><td>= another.</td></tr> +<tr><td>3.</td><td>ibutu</td><td>= beyond.</td></tr> +<tr><td>4.</td><td>ibutu-edoyoyoi</td><td>= beyond 1, or 3-1.</td></tr> +<tr><td>5.</td><td>ru-mocoso</td><td>= hand.</td></tr> +</table> + +<p>The Kamilaroi scale, given as an example of binary +formation, is partly ternary; and its word for 6, <i>guliba +guliba</i>, 3-3, is purely ternary. An occasional ternary +trace is also found in number systems otherwise decimal +or quinary vigesimal; as the <i>dlkunoutl</i>, second 3, of the +Haida Indians of British Columbia. The Karens of +India<a id="FNA-196" href="#FN-196"><sup>196</sup></a> in a system otherwise strictly decimal, exhibit +the following binary-ternary-quaternary vagary:</p> + +<table class="scale"> +<tr><td>6.</td><td>then tho</td><td>= 3 × 2.</td></tr> +<tr><td>7.</td><td>then tho ta</td><td>= 3 × 2-1.</td></tr> +<tr><td>8.</td><td>lwie tho</td><td>= 4 × 2.</td></tr> +<tr><td>9.</td><td>lwie tho ta</td><td>= 4 × 2-1.</td></tr> +</table> + +<p class="noindent">In the Wokka dialect,<a id="FNA-197" href="#FN-197"><sup>197</sup></a> found on the Burnett River, +Australia, a single ternary numeral is found, thus:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">karboon.</td></tr> +<tr><td>2.</td><td colspan="2">wombura.</td></tr> +<tr><td>3.</td><td colspan="2">chrommunda.</td></tr> +<tr><td>4.</td><td>chrommuda karboon</td><td>= 3-1.</td></tr> +</table> + +<p>Instances of quaternary numeration are less rare than +are those of ternary, and there is reason to believe that +<a class="pgnum" id="page-113" title="- 113 -"></a>this method of counting has been practised more extensively +than any other, except the binary and the three +natural methods, the quinary, the decimal, and the +vigesimal. The number of fingers on one hand is, +excluding the thumb, four. Possibly there have been +tribes among which counting by fours arose as a legitimate, +though unusual, result of finger counting; just +as there are, now and then, individuals who count on +their fingers with the forefinger as a starting-point. +But no such practice has ever been observed among +savages, and such theorizing is the merest guess-work. +Still a definite tendency to count by fours is sometimes +met with, whatever be its origin. Quaternary traces +are repeatedly to be found among the Indian languages +of British Columbia. In describing the Columbians, +Bancroft says: “Systems of numeration are simple, proceeding +by fours, fives, or tens, according to the different +languages.…”<a id="FNA-198" href="#FN-198"><sup>198</sup></a> The same preference for four is said +to have existed in primitive times in the languages of +Central Asia, and that this form of numeration, resulting +in scores of 16 and 64, was a development of finger +counting.<a id="FNA-199" href="#FN-199"><sup>199</sup></a></p> + +<p>In the Hawaiian and a few other languages of the +islands of the central Pacific, where in general the number +systems employed are decimal, we find a most interesting +case of the development, within number scales +<a class="pgnum" id="page-114" title="- 114 -"></a>already well established, of both binary and quaternary +systems. Their origin seems to have been perfectly +natural, but the systems themselves must have been +perfected very slowly. In Tahitian, Rarotongan, Mangarevan, +and other dialects found in the neighbouring +islands of those southern latitudes, certain of the higher +units, <i>tekau</i>, <i>rau</i>, <i>mano</i>, which originally signified 10, 100, +1000, have become doubled in value, and now stand for +20, 200, 2000. In Hawaiian and other dialects they have +again been doubled, and there they stand for 40, 400, +4000.<a id="FNA-200" href="#FN-200"><sup>200</sup></a> In the Marquesas group both forms are found, +the former in the southern, the latter in the northern, +part of the archipelago; and it seems probable that one +or both of these methods of numeration are scattered +somewhat widely throughout that region. The origin +of these methods is probably to be found in the fact +that, after the migration from the west toward the east, +nearly all the objects the natives would ever count in +any great numbers were small,—as yams, cocoanuts, +fish, etc.,—and would be most conveniently counted +by pairs. Hence the native, as he counted one pair, +two pairs, etc., might readily say <i>one</i>, <i>two</i>, and so on, +omitting the word “pair” altogether. Having much more +frequent occasion to employ this secondary than the +primary meaning of his numerals, the native would easily +allow the original significations to fall into disuse, and +in the lapse of time to be entirely forgotten. With a +subsequent migration to the northward a second duplication +<a class="pgnum" id="page-115" title="- 115 -"></a>might take place, and so produce the singular +effect of giving to the same numeral word three different +meanings in different parts of Oceania. To illustrate +the former or binary method of numeration, the Tahuatan, +one of the southern dialects of the Marquesas group, +may be employed.<a id="FNA-201" href="#FN-201"><sup>201</sup></a> Here the ordinary numerals are:</p> + +<table class="scale"> +<tr><td>1.</td><td>tahi<del title="should be a period">,</del><ins title="comma for period: probable printer's error">.</ins></td></tr> +<tr><td>10.</td><td>onohuu.</td></tr> +<tr><td>20.</td><td>takau.</td></tr> +<tr><td>200.</td><td>au.</td></tr> +<tr><td>2,000.</td><td>mano.</td></tr> +<tr><td>20,000.</td><td>tini.</td></tr> +<tr><td>20<ins title="obviously, 200,000 is the next step, not 20,000 again.">0</ins>,000.</td><td>tufa.</td></tr> +<tr><td>2,000,000.</td><td>pohi.</td></tr> +</table> + +<p>In counting fish, and all kinds of fruit, except breadfruit, +the scale begins with <i>tauna</i>, pair, and then, +omitting <i>onohuu</i>, they employ the same words again, +but in a modified sense. <i>Takau</i> becomes 10, <i>au</i> 100, +etc.; but as the word “pair” is understood in each case, +the value is the same as before. The table formed on +this basis would be:</p> + +<table class="units"> +<tr><td class="indent">2 (units)</td><td>= 1 tauna</td><td>= 2.</td></tr> +<tr><td>10 tauna</td><td>= 1 takau</td><td>= 20.</td></tr> +<tr><td>10 takau</td><td>= 1 au</td><td>= 200.</td></tr> +<tr><td>10 au</td><td>= 1 mano</td><td>= 2000.</td></tr> +<tr><td>10 mano</td><td>= 1 tini</td><td>= 20,000.</td></tr> +<tr><td>10 tini</td><td>= 1 tufa</td><td>= 200,000.</td></tr> +<tr><td>10 tufa</td><td>= 1 pohi</td><td>= 2,000,000.</td></tr> +</table> + +<p>For counting breadfruit they use <i>pona</i>, knot, as their +unit, breadfruit usually being tied up in knots of +four. <i>Takau</i> now takes its third signification, 40, and +becomes the base of their breadfruit system, so to +speak. For some unknown reason the next unit, 400, +is expressed by <i>tauau</i>, while <i>au</i>, which is the term that +<a class="pgnum" id="page-116" title="- 116 -"></a>would regularly stand for that number, has, by a second +duplication, come to signify 800. The next unit, <i>mano</i>, +has in a similar manner been twisted out of its original +sense, and in counting breadfruit is made to serve for +8000. In the northern, or Nukuhivan Islands, the +decimal-quaternary system is more regular. It is in +the counting of breadfruit only,<a id="FNA-202" href="#FN-202"><sup>202</sup></a></p> + +<table class="units"> +<tr><td class="indent">4 breadfruits</td><td>= 1 pona</td><td>= 4.</td></tr> +<tr><td>10 pona</td><td>= 1 toha</td><td>= 40.</td></tr> +<tr><td>10 toha</td><td>= 1 au</td><td>= 400.</td></tr> +<tr><td>10 au</td><td>= 1 mano</td><td>= 4000.</td></tr> +<tr><td>10 mano</td><td>= 1 tini</td><td>= 40,000.</td></tr> +<tr><td>10 tini</td><td>= 1 tufa</td><td>= 400,000.</td></tr> +<tr><td>10 tufa</td><td>= 1 pohi</td><td>= 4,000,000.</td></tr> +</table> + +<p>In the Hawaiian dialect this scale is, with slight +modification, the universal scale, used not only in +counting breadfruit, but any other objects as well. +The result is a complete decimal-quaternary system, +such as is found nowhere else in the world except in +this and a few of the neighbouring dialects of the +Pacific. This scale, which is almost identical with the +Nukuhivan, is<a id="FNA-203" href="#FN-203"><sup>203</sup></a></p> + +<table class="units"> +<tr><td class="indent">4 units</td><td>= 1 ha or tauna</td><td>= 4.</td></tr> +<tr><td>10 tauna</td><td>= 1 tanaha</td><td>= 40.</td></tr> +<tr><td>10 tanaha</td><td>= 1 lau</td><td>= 400.</td></tr> +<tr><td>10 lau</td><td>= 1 mano</td><td>= 4000.</td></tr> +<tr><td>10 mano</td><td>= 1 tini</td><td>= 40,000.</td></tr> +<tr><td>10 tini</td><td>= 1 lehu</td><td>= 400,000.</td></tr> +</table> + +<p><a class="pgnum" id="page-117" title="- 117 -"></a>The quaternary element thus introduced has modified +the entire structure of the Hawaiian number system. +Fifty is <i>tanaha me ta umi</i>, 40 + 10; 76 is 40 + 20 + 10 + + 6; 100 is <i>ua tanaha ma tekau</i>, 2 × 40 + 10; 200 is +<i>lima tanaha</i>, 5 × 40; and 864,895 is 2 × 400,000 + 40,000 + +6 × 4000 + 2 × 400 + 2 × 40 + 10 + 5.<a id="FNA-204" href="#FN-204"><sup>204</sup></a> Such examples show +that this secondary influence, entering and incorporating +itself as a part of a well-developed decimal system, +has radically changed it by the establishment of 4 as +the primary number base. The role which 10 now +plays is peculiar. In the natural formation of a +quaternary scale new units would be introduced at 16, +64, 256, etc.; that is, at the square, the cube, and each +successive power of the base. But, instead of this, the +new units are introduced at 10 × 4, 100 × 4, 1000 × 4, +etc.; that is, at the products of 4 by each successive +power of the old base. This leaves the scale a decimal +scale still, even while it may justly be called quaternary; +and produces one of the most singular and interesting +instances of number-system formation that has +ever been observed. In this connection it is worth +noting that these Pacific island number scales have +been developed to very high limits—in some cases into +the millions. The numerals for these large numbers +do not seem in any way indefinite, but rather to convey +to the mind of the native an idea as clear as can well +be conveyed by numbers of such magnitude. Beyond +the limits given, the islanders have indefinite expressions, +<a class="pgnum" id="page-118" title="- 118 -"></a>but as far as can be ascertained these are only used +when the limits given above have actually been passed. +To quote one more example, the Hervey Islanders, who +have a binary-decimal scale, count as follows:</p> + +<table class="units"> +<tr><td class="indent">5 kaviri (bunches of cocoanuts)</td><td>= 1 takau</td><td>= 20.</td></tr> +<tr><td>10 takau</td><td>= 1 rau</td><td>= 200.</td></tr> +<tr><td>10 rau</td><td>= 1 mano</td><td>= 2000.</td></tr> +<tr><td>10 mano</td><td>= 1 kiu</td><td>= 20,000.</td></tr> +<tr><td>10 kiu</td><td>= 1 tini</td><td>= 200,000.</td></tr> +</table> + +<p>Anything above this they speak of in an uncertain +way, as <i>mano mano</i> or <i>tini tini</i>, which may, perhaps, +be paralleled by our English phrases “myriads upon +myriads,” and “millions of millions.”<a id="FNA-205" href="#FN-205"><sup>205</sup></a> It is most remarkable +that the same quarter of the globe should +present us with the stunted number sense of the +Australians, and, side by side with it, so extended and +intelligent an appreciation of numerical values as that +possessed by many of the lesser tribes of Polynesia.</p> + +<p>The Luli of Paraguay<a id="FNA-206" href="#FN-206"><sup>206</sup></a> show a decided preference +for the base 4. This preference gives way only when +they reach the number 10, which is an ordinary digit +numeral. All numbers above that point belong rather +to decimal than to quaternary numeration. Their numerals +are:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">alapea.</td></tr> +<tr><td>2.</td><td colspan="2">tamop.</td></tr> +<tr><td>3.</td><td colspan="2">tamlip.</td></tr> +<tr><td>4.</td><td colspan="2">lokep.</td></tr> +<tr><td>5.</td><td>lokep moile alapea</td><td>= 4 with 1,</td></tr> +<tr><td></td><td class="alt">or is-alapea</td><td>= hand 1.</td></tr> +<tr><td>6.</td><td>lokep moile tamop</td><td>= 4 with 2.</td></tr> +<tr><td><a class="pgnum" id="page-119" title="- 119 -"></a>7.</td><td>lokep moile tamlip</td><td>= 4 with 3.</td></tr> +<tr><td>8.</td><td>lokep moile lokep</td><td>= 4 with 4.</td></tr> +<tr><td>9.</td><td>lokep moile lokep alapea</td><td>= 4 with 4-1.</td></tr> +<tr><td>10.</td><td>is yaoum</td><td>= all the fingers of hand.</td></tr> +<tr><td>11.</td><td>is yaoum moile alapea</td><td>= all the fingers of hand with 1.</td></tr> +<tr><td>20.</td><td>is elu yaoum</td><td>= all the fingers of hand and foot.</td></tr> +<tr><td>30.</td><td colspan="2">is elu yaoum moile is-yaoum = all the fingers of hand and foot with all the fingers of hand.</td></tr> +</table> + + +<p>Still another instance of quaternary counting, this +time carrying with it a suggestion of binary influence, +is furnished by the Mocobi<a id="FNA-207" href="#FN-207"><sup>207</sup></a> of the Parana region. +Their scale is exceedingly rude, and they use the fingers +and toes almost exclusively in counting; only +using their spoken numerals when, for any reason, they +wish to dispense with the aid of their hands and feet. +Their first eight numerals are:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">iniateda.</td></tr> +<tr><td>2.</td><td colspan="2">inabaca.</td></tr> +<tr><td>3.</td><td>inabacao caini</td><td>= 2 above.</td></tr> +<tr><td>4.</td><td>inabacao cainiba</td><td>= 2 above 2;</td></tr> +<tr><td></td><td class="alt" colspan="2">or natolatata.</td></tr> +<tr><td>5.</td><td>inibacao cainiba iniateda</td><td>= 2 above 2-1;</td></tr> +<tr><td></td><td class="alt">or natolatata iniateda</td><td>= 4-1.</td></tr> +<tr><td>6.</td><td>natolatatata inibaca</td><td>= 4-2.</td></tr> +<tr><td>7.</td><td>natolata inibacao-caini</td><td>= 4-2 above.</td></tr> +<tr><td>8.</td><td>natolata-natolata</td><td>= 4-4.</td></tr> +</table> + +<p>There is probably no recorded instance of a number +system formed on 6, 7, 8, or 9 as a base. No natural +reason exists for the choice of any of these numbers +for such a purpose; and it is hardly conceivable that +any race should proceed beyond the unintelligent +binary or quaternary stage, and then begin the formation +<a class="pgnum" id="page-120" title="- 120 -"></a>of a scale for counting with any other base than +one of the three natural bases to which allusion has +already been made. Now and then some anomalous +fragment is found imbedded in an otherwise regular +system, which carries us back to the time when the +savage was groping his way onward in his attempt to +give expression to some number greater than any he +had ever used before; and now and then one of these +fragments is such as to lead us to the border land of +the might-have-been, and to cause us to speculate on +the possibility of so great a numerical curiosity as a +senary or a septenary scale. The Bretons call 18 <i>triouec'h</i>, +3-6, but otherwise their language contains no hint +of counting by sixes; and we are left at perfect liberty +to theorize at will on the existence of so unusual +a number word. Pott remarks<a id="FNA-208" href="#FN-208"><sup>208</sup></a> that the Bolans, of +western Africa, appear to make some use of 6 as their +number base, but their system, taken as a whole, is +really a quinary-decimal. The language of the Sundas,<a id="FNA-209" href="#FN-209"><sup>209</sup></a> +or mountaineers of Java, contains traces of senary counting. +The Akra words for 7 and 8, <i>paggu</i> and <i>paniu</i>, +appear to mean 6-1 and 7-1, respectively; and the same +is true of the corresponding Tambi words <i>pagu</i> and +<i>panjo</i>.<a id="FNA-210" href="#FN-210"><sup>210</sup></a> The Watji tribe<a id="FNA-211" href="#FN-211"><sup>211</sup></a> call 6 <i>andee</i>, and 7 <i>anderee</i>, +which probably means 6-1. These words are to be +regarded as accidental variations on the ordinary laws +<a class="pgnum" id="page-121" title="- 121 -"></a>of formation, and are no more significant of a desire +to count by sixes than is the Wallachian term <i>deu-maw</i>, +which expresses 18 as 2-9, indicates the existence of a +scale of which 9 is the base. One remarkably interesting +number system is that exhibited by the Mosquito +tribe<a id="FNA-212" href="#FN-212"><sup>212</sup></a> of Central America, who possess an extensive +quinary-vigesimal scale containing one binary and +three senary compounds. The first ten words of this +singular scale, which has already been quoted, are:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">kumi.</td></tr> +<tr><td>2.</td><td>wal.</td></tr> +<tr><td>3.</td><td colspan="2">niupa.</td></tr> +<tr><td>4.</td><td>wal-wal</td><td>= 2-2.</td></tr> +<tr><td>5.</td><td>mata-sip</td><td>= fingers of one hand.</td></tr> +<tr><td>6.</td><td colspan="2">matlalkabe.</td></tr> +<tr><td>7.</td><td>matlalkabe pura kumi</td><td>= 6 + 1.</td></tr> +<tr><td>8.</td><td>matlalkabe pura wal</td><td>= 6 + 2.</td></tr> +<tr><td>9.</td><td>matlalkabe pura niupa</td><td>= 6 + 3.</td></tr> +<tr><td>10.</td><td>mata-wal-sip</td><td>= fingers of the second hand.</td></tr> +</table> + +<p class="noindent">In passing from 6 to 7, this tribe, also, has varied the +almost universal law of progression, and has called 7 +6-1. Their 8 and 9 are formed in a similar manner; +but at 10 the ordinary method is resumed, and is continued +from that point onward. Few number systems +contain as many as three numerals which are associated +with 6 as their base. In nearly all instances we find +such numerals singly, or at most in pairs; and in the +structure of any system as a whole, they are of no importance +whatever. For example, in the Pawnee, a pure +decimal scale, we find the following odd sequence:<a id="FNA-213" href="#FN-213"><sup>213</sup></a></p> + +<table class="scale"> +<tr><td><a class="pgnum" id="page-122" title="- 122 -"></a>6.</td><td colspan="2">shekshabish.</td></tr> +<tr><td>7.</td><td>petkoshekshabish</td><td>= 2-6, <i>i.e.</i> 2d 6.</td></tr> +<tr><td>8.</td><td>touwetshabish</td><td>= 3-6, <i>i.e.</i> 3d 6.</td></tr> +<tr><td>9.</td><td>loksherewa</td><td>= 10 − 1.</td></tr> +</table> + +<p>In the Uainuma scale the expressions for 7 and 8 are +obviously referred to 6, though the meaning of 7 is +not given, and it is impossible to guess what it really +does signify. The numerals in question are:<a id="FNA-214" href="#FN-214"><sup>214</sup></a></p> + +<table class="scale"> +<tr><td>6.</td><td colspan="2">aira-ettagapi.</td></tr> +<tr><td>7.</td><td colspan="2">aira-ettagapi-hairiwigani-apecapecapsi.</td></tr> +<tr><td>8.</td><td>aira-ettagapi-matschahma</td><td>= 6 + 2.</td></tr> +</table> + +<p>In the dialect of the Mille tribe a single trace of +senary counting appears, as the numerals given below +show:<a id="FNA-215" href="#FN-215"><sup>215</sup></a></p> + +<table class="scale"> +<tr><td>6.</td><td colspan="2">dildjidji.</td></tr> +<tr><td>7.</td><td>dildjidji me djuun</td><td>= 6 + 1.</td></tr> +</table> + +<p>Finally, in the numerals used by the natives of the +Marshall Islands, the following curiously irregular sequence +also contains a single senary numeral:<a id="FNA-216" href="#FN-216"><sup>216</sup></a></p> + +<table class="scale"> +<tr><td>6.</td><td>thil thino</td><td>= 3 + 3.</td></tr> +<tr><td>7.</td><td>thilthilim-thuon</td><td>= 6 + 1.</td></tr> +<tr><td>8.</td><td>rua-li-dok</td><td>= 10 − 2.</td></tr> +<tr><td>9.</td><td>ruathim-thuon</td><td>= 10 − 2 + 1.</td></tr> +</table> + +<p>Many years ago a statement appeared which at once +attracted attention and awakened curiosity. It was to +the effect that the Maoris, the aboriginal inhabitants of +<a class="pgnum" id="page-123" title="- 123 -"></a>New Zealand, used as the basis of their numeral system +the number 11; and that the system was quite +extensively developed, having simple words for 121 +and 1331, <i>i.e.</i> for the square and cube of 11. No apparent +reason existed for this anomaly, and the Maori +scale was for a long time looked upon as something +quite exceptional and outside all ordinary rules of +number-system formation. But a closer and more accurate +knowledge of the Maori language and customs +served to correct the mistake, and to show that this +system was a simple decimal system, and that the error +arose from the following habit. Sometimes when counting +a number of objects the Maoris would put aside 1 +to represent each 10, and then those so set aside would +afterward be counted to ascertain the number of tens +in the heap. Early observers among this people, seeing +them count 10 and then set aside 1, at the same time +pronouncing the word <i>tekau</i>, imagined that this word +meant 11, and that the ignorant savage was making +use of this number as his base. This misconception +found its way into the early New Zealand dictionary, +but was corrected in later editions. It is here mentioned +only because of the wide diffusion of the error, +and the interest it has always excited.<a id="FNA-217" href="#FN-217"><sup>217</sup></a></p> + +<p>Aside from our common decimal scale, there exist in +the English language other methods of counting, some +of them formal enough to be dignified by the term +<a class="pgnum" id="page-124" title="- 124 -"></a><i>system</i>—as the sexagesimal method of measuring time +and angular magnitude; and the duodecimal system of +reckoning, so extensively used in buying and selling. +Of these systems, other than decimal, two are noticed +by Tylor,<a id="FNA-218" href="#FN-218"><sup>218</sup></a> and commented on at some length, as +follows:</p> + +<p>“One is the well-known dicing set, <i>ace</i>, <i>deuce</i>, <i>tray</i>, +<i>cater</i>, <i>cinque</i>, <i>size</i>; thus <i>size-ace</i> is 6-1, <i>cinques</i> or <i>sinks</i>, +double 5. These came to us from France, and correspond +with the common French numerals, except <i>ace</i>, +which is Latin <i>as</i>, a word of great philological interest, +meaning ‘one.’ The other borrowed set is to be found +in the <i>Slang Dictionary</i>. It appears that the English +street-folk have adopted as a means of secret communication +a set of Italian numerals from the organ-grinders +and image-sellers, or by other ways through which +Italian or Lingua Franca is brought into the low +neighbourhoods of London. In so doing they have performed +a philological operation not only curious but +instructive. By copying such expressions as <i>due soldi</i>, +<i>tre soldi</i>, as equivalent to ‘twopence,’ ‘threepence,’ the +word <i>saltee</i> became a recognized slang term for ‘penny’; +and pence are reckoned as follows:</p> + +<table id="saltees"> +<tr><td>oney saltee</td><td>1d.</td><td>uno soldo.</td></tr> +<tr><td>dooe saltee</td><td>2d.</td><td>due soldi.</td></tr> +<tr><td>tray saltee</td><td>3d.</td><td>tre soldi.</td></tr> +<tr><td>quarterer saltee</td><td>4d.</td><td>quattro soldi.</td></tr> +<tr><td>chinker saltee</td><td>5d.</td><td>cinque soldi.</td></tr> +<tr><td><a class="pgnum" id="page-125" title="- 125 -"></a>say saltee</td><td>6d.</td><td>sei soldi.</td></tr> +<tr><td>say oney saltee, or setter saltee</td><td>7d.</td><td>sette soldi.</td></tr> +<tr><td>say dooe saltee, or otter saltee</td><td>8d.</td><td>otto soldi.</td></tr> +<tr><td>say tray saltee, or nobba saltee</td><td>9d.</td><td>nove soldi.</td></tr> +<tr><td>say quarterer saltee, or dacha saltee</td><td>10d.</td><td>dieci soldi.</td></tr> +<tr><td>say chinker saltee or dacha oney saltee</td><td>11d.</td><td>undici soldi.</td></tr> +<tr><td>oney beong</td><td>1s.</td></tr> +<tr><td>a beong say saltee</td><td>1s. 6d.</td></tr> +<tr><td>dooe beong say saltee, or madza caroon</td><td>2s. 6d.</td><td>(half-crown, mezza corona).</td></tr> +</table> + + +<p class="noindent">One of these series simply adopts Italian numerals +decimally. But the other, when it has reached 6, +having had enough of novelty, makes 7 by 6-1, and so +forth. It is for no abstract reason that 6 is thus made +the turning-point, but simply because the costermonger is +adding pence up to the silver sixpence, and then adding +pence again up to the shilling. Thus our duodecimal +coinage has led to the practice of counting by sixes, and +produced a philological curiosity, a real senary notation.”</p> + +<p>In addition to the two methods of counting here +alluded to, another may be mentioned, which is equally +instructive as showing how readily any special method +of reckoning may be developed out of the needs arising +in connection with any special line of work. As +is well known, it is the custom in ocean, lake, and +river navigation to measure soundings by the fathom. +On the Mississippi River, where constant vigilance is +needed because of the rapid shifting of sand-bars, a +special sounding nomenclature has come into vogue,<a id="FNA-219" href="#FN-219"><sup>219</sup></a> +which the following terms will illustrate:</p> + +<table class="units"> +<tr><td class="indent"><a class="pgnum" id="page-126" title="- 126 -"></a>5</td><td>ft.</td><td>= five feet.</td></tr> +<tr><td class="indent">6</td><td>ft.</td><td>= six feet.</td></tr> +<tr><td class="indent">9</td><td>ft.</td><td>= nine feet.</td></tr> +<tr><td>10-1/2</td><td>ft.</td><td>= a quarter less twain; <i>i.e.</i> a quarter of a fathom less than 2.</td></tr> +<tr><td>12</td><td>ft.</td><td>= mark twain.</td></tr> +<tr><td>13-1/2</td><td>ft.</td><td>= a quarter twain.</td></tr> +<tr><td>16-1/2</td><td>ft.</td><td>= a quarter less three.</td></tr> +<tr><td>18</td><td>ft.</td><td>= mark three.</td></tr> +<tr><td>19-1/2</td><td>ft.</td><td>= a quarter three.</td></tr> +<tr><td>24</td><td>ft.</td><td>= deep four.</td></tr> +</table> + +<p>As the soundings are taken, the readings are called +off in the manner indicated in the table; 10-1/2 feet +being “a quarter less twain,” 12 feet “mark twain,” +etc. Any sounding above “deep four” is reported as +“no bottom.” In the Atlantic and Gulf waters on the +coast of this country the same system prevails, only it +is extended to meet the requirements of the deeper +soundings there found, and instead of “six feet,” “mark +twain,” etc., we find the fuller expressions, “by the +mark one,” “by the mark two,” and so on, as far as +the depth requires. This example also suggests the +older and far more widely diffused method of reckoning +time at sea by bells; a system in which “one +bell,” “two bells,” “three bells,” etc., mark the passage +of time for the sailor as distinctly as the hands of the +clock could do it. Other examples of a similar nature +will readily suggest themselves to the mind.</p> + +<p>Two possible number systems that have, for purely +theoretical reasons, attracted much attention, are the +octonary and the duodecimal systems. In favour of the +octonary system it is urged that 8 is an exact power +of 2; or in other words, a large number of repeated +halves can be taken with 8 as a starting-point, without +<a class="pgnum" id="page-127" title="- 127 -"></a>producing a fractional result. With 8 as a base we +should obtain by successive halvings, 4, 2, 1. A similar +process in our decimal scale gives 5, 2-1/2, 1-1/4. All +this is undeniably true, but, granting the argument up +to this point, one is then tempted to ask “What of +it?” A certain degree of simplicity would thereby be +introduced into the Theory of Numbers; but the only +persons sufficiently interested in this branch of mathematics +to appreciate the benefit thus obtained are +already trained mathematicians, who are concerned +rather with the pure science involved, than with reckoning +on any special base. A slightly increased simplicity +would appear in the work of stockbrokers, and +others who reckon extensively by quarters, eighths, and +sixteenths. But such men experience no difficulty whatever +in performing their mental computations in the +decimal system; and they acquire through constant +practice such quickness and accuracy of calculation, +that it is difficult to see how octonary reckoning would +materially assist them. Altogether, the reasons that +have in the past been adduced in favour of this form of +arithmetic seem trivial. There is no record of any +tribe that ever counted by eights, nor is there the +slightest likelihood that such a system could ever meet +with any general favour. It is said that the ancient +Saxons used the octonary system,<a id="FNA-220" href="#FN-220"><sup>220</sup></a> but how, or for +what purposes, is not stated. It is not to be supposed +<a class="pgnum" id="page-128" title="- 128 -"></a>that this was the common system of counting, for it is +well known that the decimal scale was in use as far +back as the evidence of language will take us. But +the field of speculation into which one is led by the +octonary scale has proved most attractive to some, and +the conclusion has been soberly reached, that in the +history of the Aryan race the octonary was to be regarded +as the predecessor of the decimal scale. In +support of this theory no direct evidence is brought +forward, but certain verbal resemblances. Those ignes +fatuii of the philologist are made to perform the duty +of supporting an hypothesis which would never have +existed but for their own treacherous suggestions. +Here is one of the most attractive of them:</p> + +<p>Between the Latin words <i>novus</i>, new, and <i>novem</i>, nine, +there exists a resemblance so close that it may well be +more than accidental. Nine is, then, the <i>new</i> number; +that is, the first number on a new count, of which 8 +must originally have been the base. Pursuing this +thought by investigation into different languages, the +same resemblance is found there. Hence the theory is +strengthened by corroborative evidence. In language +after language the same resemblance is found, until it +seems impossible to doubt, that in prehistoric times, 9 +<i>was</i> the new number—the beginning of a second tale. +The following table will show how widely spread is +this coincidence:</p> + +<table id="nine-new"> +<tr><td>Sanskrit, navan</td><td>= 9.</td><td>nava</td><td>= new.</td></tr> +<tr><td>Persian, nuh</td><td>= 9.</td><td>nau</td><td>= new.</td></tr> +<tr><td><a class="pgnum" id="page-129" title="- 129 -"></a>Greek, <span class="greek">ἐννέα</span></td><td>= 9.</td><td><span class="greek">νέος</span></td><td>= new.</td></tr> +<tr><td>Latin, novem</td><td>= 9.</td><td>novus</td><td>= new.</td></tr> +<tr><td>German, neun</td><td>= 9.</td><td>neu</td><td>= new.</td></tr> +<tr><td>Swedish, nio</td><td>= 9.</td><td>ny</td><td>= new.</td></tr> +<tr><td>Dutch, negen</td><td>= 9.</td><td>nieuw</td><td>= new.</td></tr> +<tr><td>Danish, ni</td><td>= 9.</td><td>ny</td><td>= new.</td></tr> +<tr><td>Icelandic, nyr</td><td>= 9.</td><td>niu</td><td>= new.</td></tr> +<tr><td>English, nine</td><td>= 9.</td><td>new</td><td>= new.</td></tr> +<tr><td>French, neuf</td><td>= 9.</td><td>nouveau</td><td>= new.</td></tr> +<tr><td>Spanish, nueve</td><td>= 9.</td><td>neuvo</td><td>= new.</td></tr> +<tr><td>Italian, nove</td><td>= 9.</td><td>nuovo</td><td>= new.</td></tr> +<tr><td>Portuguese, nove</td><td>= 9.</td><td>novo</td><td>= new.</td></tr> +<tr><td>Irish, naoi</td><td>= 9.</td><td>nus</td><td>= new.</td></tr> +<tr><td>Welsh, naw</td><td>= 9.</td><td>newydd</td><td>= new.</td></tr> +<tr><td>Breton, nevez</td><td>= 9.</td><td>nuhue</td><td>= new.<a id="FNA-221" href="#FN-221"><sup>221</sup></a></td></tr> +</table> + + +<p>This table might be extended still further, but the +above examples show how widely diffused throughout +the Aryan languages is this resemblance. The list +certainly is an impressive one, and the student is at +first thought tempted to ask whether all these resemblances +can possibly have been accidental. But a single +consideration sweeps away the entire argument as +though it were a cobweb. All the languages through +which this verbal likeness runs are derived directly +or indirectly from one common stock; and the common +every-day words, “nine” and “new,” have been transmitted +from that primitive tongue into all these linguistic +offspring with but little change. Not only are +the two words in question akin in each individual language, +but <i>they are akin in all the languages</i>. Hence +<a class="pgnum" id="page-130" title="- 130 -"></a>all these resemblances reduce to a single resemblance, +or perhaps identity, that between the Aryan words for +“nine” and “new.” This was probably an accidental +resemblance, no more significant than any one of the +scores of other similar cases occurring in every language. +If there were any further evidence of the +former existence of an Aryan octonary scale, the coincidence +would possess a certain degree of significance; +but not a shred has ever been produced which is +worthy of consideration. If our remote ancestors ever +counted by eights, we are entirely ignorant of the fact, +and must remain so until much more is known of their +language than scholars now have at their command. +The word resemblances noted above are hardly more +significant than those occurring in two Polynesian languages, +the Fatuhivan and the Nakuhivan,<a id="FNA-222" href="#FN-222"><sup>222</sup></a> where +“new” is associated with the number 7. In the former +case 7 is <i>fitu</i>, and “new” is <i>fou</i>; in the latter 7 is +<i>hitu</i>, and “new” is <i>hou</i>. But no one has, because of this +likeness, ever suggested that these tribes ever counted +by the senary method. Another equally trivial resemblance +occurs in the Tawgy and the Kamassin languages,<a id="FNA-223" href="#FN-223"><sup>223</sup></a> +thus:</p> + +<div class="lefthalf"> +<table class="scale"> +<caption>Tawgy.</caption> +<tr><td>8.</td><td>siti-data</td><td>= 2 × 4.</td></tr> +<tr><td>9.</td><td>nameaitjuma</td><td>= another.</td></tr> +</table> +</div> +<div class="righthalf"> +<table class="scale"> +<caption>Kamassin.</caption> +<tr><td>8.</td><td>sin-the'de</td><td>= 2 × 4.</td></tr> +<tr><td>9.</td><td>amithun</td><td>= another.</td></tr> +</table> +</div> + +<p><a class="pgnum" id="page-131" title="- 131 -"></a>But it would be childish to argue, from this fact +alone, that either 4 or 8 was the number base used.</p> + +<p>In a recent antiquarian work of considerable interest, +the author examines into the question of a former +octonary system of counting among the various races +of the world, particularly those of Asia, and brings to +light much curious and entertaining material respecting +the use of this number. Its use and importance in +China, India, and central Asia, as well as among some +of the islands of the Pacific, and in Central America, +leads him to the conclusion that there was a time, long +before the beginning of recorded history, when 8 was +the common number base of the world. But his conclusion +has no basis in his own material even. The +argument cannot be examined here, but any one who +cares to investigate it can find there an excellent illustration +of the fact that a pet theory may take complete +possession of its originator, and reduce him finally to a +state of infantile subjugation.<a id="FNA-224" href="#FN-224"><sup>224</sup></a></p> + +<p>Of all numbers upon which a system could be based, +12 seems to combine in itself the greatest number of +advantages. It is capable of division by 2, 3, 4, and 6, +and hence admits of the taking of halves, thirds, quarters, +and sixths of itself without the introduction of +fractions in the result. From a commercial stand-point +this advantage is very great; so great that many have +seriously advocated the entire abolition of the decimal +<a class="pgnum" id="page-132" title="- 132 -"></a>scale, and the substitution of the duodecimal in its +stead. It is said that Charles XII. of Sweden was +actually contemplating such a change in his dominions +at the time of his death. In pursuance of this idea, +some writers have gone so far as to suggest symbols +for 10 and 11, and to recast our entire numeral nomenclature +to conform to the duodecimal base.<a id="FNA-225" href="#FN-225"><sup>225</sup></a> Were such +a change made, we should express the first nine numbers +as at present, 10 and 11 by new, single symbols, +and 12 by 10. From this point the progression would +be regular, as in the decimal scale—only the same +combination of figures in the different scales would +mean very different things. Thus, 17 in the decimal +scale would become 15 in the duodecimal; 144 in the +decimal would become 100 in the duodecimal; and +1728, the cube of the new base, would of course be +represented by the figures 1000.</p> + +<p>It is impossible that any such change can ever meet +with general or even partial favour, so firmly has the +decimal scale become intrenched in its position. But it +is more than probable that a large part of the world of +trade and commerce will continue to buy and sell by the +dozen, the gross, or some multiple or fraction of the one +or the other, as long as buying and selling shall continue. +Such has been its custom for centuries, and such +will doubtless be its custom for centuries to come. The +duodecimal is not a natural scale in the same sense as +<a class="pgnum" id="page-133" title="- 133 -"></a>are the quinary, the decimal, and the vigesimal; but it +is a system which is called into being long after the +complete development of one of the natural systems, +solely because of the simple and familiar fractions into +which its base is divided. It is the scale of civilization, +just as the three common scales are the scales of nature. +But an example of its use was long sought for in vain +among the primitive races of the world. Humboldt, in +commenting on the number systems of the various peoples +he had visited during his travels, remarked that no race +had ever used exclusively that best of bases, 12. But +it has recently been announced<a id="FNA-226" href="#FN-226"><sup>226</sup></a> that the discovery of +such a tribe had actually been made, and that the +Aphos of Benuë, an African tribe, count to 12 by +simple words, and then for 13 say 12-1, for 14, 12-2, +etc. This report has yet to be verified, but if true +it will constitute a most interesting addition to anthropological +knowledge.</p> + + + + + +<h2><a class="pgnum" id="page-134" title="- 134 -"></a>Chapter VI.</h2> +<h3>The Quinary System.</h3> + + +<p><span class="initial">The</span> origin of the quinary mode of counting has been +discussed with some fulness in a preceding chapter, +and upon that question but little more need be said. +It is the first of the natural systems. When the savage +has finished his count of the fingers of a single +hand, he has reached this natural number base. At +this point he ceases to use simple numbers, and begins +the process of compounding. By some one of the +numerous methods illustrated in earlier chapters, he +passes from 5 to 10, using here the fingers of his +second hand. He now has two fives; and, just as we +say “twenty,” <i>i.e.</i> two tens, he says “two hands,” +“the second hand finished,” “all the fingers,” “the +fingers of both hands,” “all the fingers come to an +end,” or, much more rarely, “one man.” That is, he +is, in one of the many ways at his command, saying +“two fives.” At 15 he has “three hands” or “one +foot”; and at 20 he pauses with “four hands,” “hands +and feet,” “both feet,” “all the fingers of hands and +feet,” “hands and feet finished,” or, more probably, +“one man.” All these modes of expression are strictly +natural, and all have been found in the number scales +<a class="pgnum" id="page-135" title="- 135 -"></a>which were, and in many cases still are, in daily use +among the uncivilized races of mankind.</p> + +<p>In its structure the quinary is the simplest, the most +primitive, of the natural systems. Its base is almost +always expressed by a word meaning “hand,” or by +some equivalent circumlocution, and its digital origin +is usually traced without difficulty. A consistent formation +would require the expression of 10 by some +phrase meaning “two fives,” 15 by “three fives,” etc. +Such a scale is the one obtained from the Betoya language, +already mentioned in Chapter III., where the formation +of the numerals is purely quinary, as the following +indicate:<a id="FNA-227" href="#FN-227"><sup>227</sup></a></p> + + +<table class="scale"> +<tr><td>5.</td><td>teente</td><td>= 1 hand.</td></tr> +<tr><td>10.</td><td>cayaente, or caya huena</td><td>= 2 hands.</td></tr> +<tr><td>15.</td><td>toazumba-ente</td><td>= 3 hands.</td></tr> +<tr><td>20.</td><td>caesa-ente</td><td>= 4 hands.</td></tr> +</table> + +<p>The same formation appears, with greater or less distinctness, +in many of the quinary scales already quoted, +and in many more of which mention might be made. +Collecting the significant numerals from a few such +scales, and tabulating them for the sake of convenience +of comparison, we see this point clearly illustrated +by the following:</p> + +<table class="scale"> +<caption>Tamanac.</caption> + +<tr><td>5.</td><td>amnaitone</td><td>= 1 hand.</td></tr> +<tr><td>10.</td><td>amna atse ponare</td><td>= 2 hands.</td></tr> +</table> + +<table class="scale"> +<caption>Arawak, Guiana.</caption> +<tr><td>5.</td><td>abba tekkabe</td><td>= 1 hand.</td></tr> +<tr><td>10.</td><td>biamantekkabe</td><td>= 2 hands.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-136" title="- 136 -"></a>Jiviro.</caption> + +<tr><td>5.</td><td>alacötegladu</td><td>= 1 hand.</td></tr> +<tr><td>10.</td><td>catögladu</td><td>= 2 hands.</td></tr> +</table> + +<table class="scale"> +<caption>Niam Niam</caption> + +<tr><td>5.</td><td colspan="2">biswe</td></tr> +<tr><td>10.</td><td>bauwe</td><td>= 2d 5.</td></tr> +</table> + +<table class="scale"> +<caption>Nengones</caption> + +<tr><td>5.</td><td>se dono</td><td>= the end (of the fingers of 1 hand).</td></tr> +<tr><td>10.</td><td>rewe tubenine</td><td>= 2 series (of fingers).</td></tr> +</table> + +<table class="scale"> +<caption>Sesake.<a id="FNA-228" href="#FN-228"><sup>228</sup></a></caption> + +<tr><td>5.</td><td>lima</td><td>= hand.</td></tr> +<tr><td>10.</td><td>dua lima</td><td>= 2 hands.</td></tr> +</table> + +<table class="scale"> +<caption>Ambrym.<a id="FNA-229" href="#FN-229"><sup>229</sup></a></caption> + +<tr><td>5.</td><td>lim</td><td>= hand.</td></tr> +<tr><td>10.</td><td>ra-lim</td><td>= 2 hands.</td></tr> +</table> + +<table class="scale"> +<caption>Pama.<a href="#FN-229"><sup>229</sup></a></caption> + +<tr><td>5.</td><td>e-lime</td><td>= hand.</td></tr> +<tr><td>10.</td><td>ha-lua-lim</td><td>= the 2 hands.</td></tr> +</table> + +<table class="scale"> +<caption>Dinka.<a id="FNA-230" href="#FN-230"><sup>230</sup></a></caption> + +<tr><td>5.</td><td colspan="2">wdyets.</td></tr> +<tr><td>10.</td><td>wtyer, or wtyar</td><td>= 5 × 2.</td></tr> +</table> + + +<table class="scale"> +<caption>Bari</caption> + +<tr><td>5.</td><td colspan="2">kanat</td></tr> +<tr><td>10.</td><td>puök</td><td>= 5 + 5?</td></tr> +</table> + +<table class="scale"> +<caption>Kanuri</caption> + +<tr><td>5.</td><td colspan="2">ugu.</td></tr> +<tr><td>10.</td><td>megu</td><td>= 2 × 5.</td></tr> +</table> + +<table class="scale"> +<caption>Rio Norte and San Antonio.<a id="FNA-231" href="#FN-231"><sup>231</sup></a></caption> + +<tr><td>5.</td><td colspan="2">juyopamauj.</td></tr> +<tr><td>10.</td><td>juyopamauj ajte</td><td>= 5 × 2.</td></tr> +</table> + +<table class="scale"> +<caption>Api.<a id="FNA-232" href="#FN-232"><sup>232</sup></a></caption> + +<tr><td>5.</td><td colspan="2">lima.</td></tr> +<tr><td>10.</td><td>lua-lima</td><td>= 2 × 5.</td></tr> +</table> + +<table class="scale"> +<caption>Erromango</caption> + +<tr><td>5.</td><td colspan="2">suku-rim.</td></tr> +<tr><td>10.</td><td>nduru-lim</td><td>= 2 × 5.</td></tr> +</table> + + +<table class="scale"> +<caption>Tlingit, British Columbia.<a id="FNA-233" href="#FN-233"><sup>233</sup></a></caption> + +<tr><td>5.</td><td colspan="2">kedjin (from djin = hand).</td></tr> +<tr><td>10.</td><td>djinkat</td><td>= both hands?</td></tr> +</table> + +<p>Thus far the quinary formation is simple and regular; +and in view of the evidence with which these and +<a class="pgnum" id="page-137" title="- 137 -"></a>similar illustrations furnish us, it is most surprising to +find an eminent authority making the unequivocal statement +that the number 10 is nowhere expressed by 2 +fives<a id="FNA-234" href="#FN-234"><sup>234</sup></a>—that all tribes which begin their count on a +quinary base express 10 by a simple word. It is a +fact, as will be fully illustrated in the following pages, +that quinary number systems, when extended, usually +merge into either the decimal or the vigesimal. The +result is, of course, a compound of two, and sometimes +of three, systems in one scale. A pure quinary or +vigesimal number system is exceedingly rare; but quinary +scales certainly do exist in which, as far as we +possess the numerals, no trace of any other influence +appears. It is also to be noticed that some tribes, like +the Eskimos of Point Barrow, though their systems may +properly be classed as mixed systems, exhibit a decided +preference for 5 as a base, and in counting objects, divided +into groups of 5, obtaining the sum in this way.<a id="FNA-235" href="#FN-235"><sup>235</sup></a></p> + +<p>But the savage, after counting up to 10, often finds +himself unconsciously impelled to depart from his strict +reckoning by fives, and to assume a new basis of reference. +Take, for example, the Zuñi system, in which +the first 2 fives are:</p> + +<table class="scale"> +<tr><td>5.</td><td>öpte</td><td>= the notched off.</td></tr> +<tr><td>10.</td><td>astem'thla</td><td>= all the fingers.</td></tr> +</table> + +<p class="noindent">It will be noticed that the Zuñi does not say “two +hands,” or “the fingers of both hands,” but simply “all +<a class="pgnum" id="page-138" title="- 138 -"></a>the fingers.” The 5 is no longer prominent, but instead +the mere notion of one entire count of the fingers has +taken its place. The division of the fingers into two sets +of five each is still in his mind, but it is no longer the +leading idea. As the count proceeds further, the quinary +base may be retained, or it may be supplanted by a decimal +or a vigesimal base. How readily the one or the +other may predominate is seen by a glance at the following +numerals:</p> + +<table class="scale"> +<caption>Galibi.<a id="FNA-236" href="#FN-236"><sup>236</sup></a></caption> + +<tr><td>5.</td><td>atoneigne oietonaï</td><td>= 1 hand.</td></tr> +<tr><td>10.</td><td>oia batoue</td><td>= the other hand.</td></tr> +<tr><td>20.</td><td>poupoupatoret oupoume</td><td>= feet and hands.</td></tr> +<tr><td>40.</td><td>opoupoume</td><td>= twice the feet and hands.</td></tr> +</table> + +<table class="scale"> +<caption>Guarani.<a id="FNA-237" href="#FN-237"><sup>237</sup></a></caption> + +<tr><td>5.</td><td>ace popetei</td><td>= 1 hand.</td></tr> +<tr><td>10.</td><td>ace pomocoi</td><td>= 2 hands.</td></tr> +<tr><td>20.</td><td>acepo acepiabe</td><td>= hands and feet.</td></tr> +</table> + +<table class="scale"> +<caption>Fate.<a id="FNA-238" href="#FN-238"><sup>238</sup></a></caption> + +<tr><td>5.</td><td>lima</td><td>= hand.</td></tr> +<tr><td>10.</td><td>relima</td><td>= 2 hands.</td></tr> +<tr><td>20.</td><td>relima rua</td><td>= (2 × 5) × 2.</td></tr> +</table> + +<table class="scale"> +<caption>Kiriri</caption> + +<tr><td>5.</td><td>mibika misa</td><td>= 1 hand.</td></tr> +<tr><td>10.</td><td>mikriba misa sai</td><td>= both hands.</td></tr> +<tr><td>20.</td><td>mikriba nusa ideko ibi sai</td><td>= both hands together with the feet.</td></tr> +</table> + +<table class="scale"> +<caption>Zamuco</caption> + +<tr><td>5.</td><td>tsuena yimana-ite</td><td>= ended 1 hand.</td></tr> +<tr><td>10.</td><td>tsuena yimana-die</td><td>= ended both hands.</td></tr> +<tr><td>20.</td><td>tsuena yiri-die</td><td>= ended both feet.</td></tr> +</table> + +<table class="scale"> +<caption>Pikumbul</caption> + +<tr><td>5.</td><td colspan="2">mulanbu.</td></tr> +<tr><td>10.</td><td>bularin murra</td><td>= belonging to the two hands.</td></tr> +<tr><td><a class="pgnum" id="page-139" title="- 139 -"></a>15.</td><td>mulanba dinna</td><td>= 5 toes added on (to the 10 fingers).</td></tr> +<tr><td>20.</td><td>bularin dinna</td><td>= belonging to the 2 feet.</td></tr> +</table> + +<table class="scale"> +<caption>Yaruros.<a id="FNA-239" href="#FN-239"><sup>239</sup></a></caption> + +<tr><td>5.</td><td>kani-iktsi-mo</td><td>= 1 hand alone.</td></tr> +<tr><td>10.</td><td>yowa-iktsi-bo</td><td>= all the hands.</td></tr> +<tr><td>15.</td><td>kani-tao-mo</td><td>= 1 foot alone.</td></tr> +<tr><td>20.</td><td>kani-pume</td><td>= 1 man.</td></tr> +</table> + +<p>By the time 20 is reached the savage has probably +allowed his conception of any aggregate to be so far +modified that this number does not present itself to +his mind as 4 fives. It may find expression in some +phraseology such as the Kiriris employ—“both hands +together with the feet”—or in the shorter “ended both +feet” of the Zamucos, in which case we may presume +that he is conscious that his count has been completed +by means of the four sets of fives which are furnished +by his hands and feet. But it is at least equally probable +that he instinctively divides his total into 2 tens, +and thus passes unconsciously from the quinary into the +decimal scale. Again, the summing up of the 10 fingers +and 10 toes often results in the concept of a single +whole, a lump sum, so to speak, and the savage then +says “one man,” or something that gives utterance to +this thought of a new unit. This leads the quinary into +the vigesimal scale, and produces the combination so +often found in certain parts of the world. Thus the +inevitable tendency of any number system of quinary +origin is toward the establishment of another and larger +base, and the formation of a number system in which +<a class="pgnum" id="page-140" title="- 140 -"></a>both are used. Wherever this is done, the greater of +the two bases is always to be regarded as the principal +number base of the language, and the 5 as entirely subordinate +to it. It is hardly correct to say that, as a +number system is extended, the quinary element disappears +and gives place to the decimal or vigesimal, +but rather that it becomes a factor of quite secondary +importance in the development of the scale. If, for +example, 8 is expressed by 5-3 in a quinary decimal +system, 98 will be 9 × 10 + 5-3. The quinary element +does not disappear, but merely sinks into a relatively +unimportant position.</p> + +<p>One of the purest examples of quinary numeration +is that furnished by the Betoya scale, already given in +full in Chapter III., and briefly mentioned at the beginning +of this chapter. In the simplicity and regularity +of its construction it is so noteworthy that it is worth +repeating, as the first of the long list of quinary +systems given in the following pages. No further +comment is needed on it than that already made in +connection with its digital significance. As far as +given by Dr. Brinton the scale is:</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">tey.</td></tr> +<tr><td>2.</td><td colspan="2">cayapa.</td></tr> +<tr><td>3.</td><td colspan="2">toazumba.</td></tr> +<tr><td>4.</td><td>cajezea</td><td>= 2 with plural termination.</td></tr> +<tr><td>5.</td><td>teente</td><td>= hand.</td></tr> +<tr><td>6.</td><td>teyente tey</td><td>= hand 1.</td></tr> +<tr><td>7.</td><td>teyente cayapa</td><td>= hand 2.</td></tr> +<tr><td>8.</td><td>teyente toazumba</td><td>= hand 3.</td></tr> +<tr><td>9.</td><td>teyente caesea</td><td>= hand 4.</td></tr> +<tr><td>10.</td><td>caya ente, or caya huena</td><td>= 2 hands.</td></tr> +<tr><td>11.</td><td>caya ente-tey</td><td>= 2 hands 1.</td></tr> +<tr><td>15.</td><td>toazumba-ente</td><td>= 3 hands.</td></tr> +<tr><td>16.</td><td>toazumba-ente-tey</td><td>= 3 hands 1.</td></tr> +<tr><td>20.</td><td>caesea ente</td><td>= 4 hands.</td></tr> +</table> + +<p><a class="pgnum" id="page-141" title="- 141 -"></a>A far more common method of progression is furnished +by languages which interrupt the quinary formation +at 10, and express that number by a single +word. Any scale in which this takes place can, from +this point onward, be quinary only in the subordinate +sense to which allusion has just been made. Examples +of this are furnished in a more or less perfect manner +by nearly all so-called quinary-vigesimal and quinary-decimal +scales. As fairly representing this phase of +number-system structure, I have selected the first 20 +numerals from the following languages:</p> + +<table class="scale"> +<caption>Welsh.<a id="FNA-240" href="#FN-240"><sup>240</sup></a></caption> +<tr><td>1.</td><td colspan="2">un.</td></tr> +<tr><td>2.</td><td colspan="2">dau.</td></tr> +<tr><td>3.</td><td colspan="2">tri.</td></tr> +<tr><td>4.</td><td colspan="2">pedwar.</td></tr> +<tr><td>5.</td><td colspan="2">pump.</td></tr> +<tr><td>6.</td><td colspan="2">chwech.</td></tr> +<tr><td>7.</td><td colspan="2">saith.</td></tr> +<tr><td>8.</td><td colspan="2">wyth.</td></tr> +<tr><td>9.</td><td colspan="2">naw.</td></tr> +<tr><td>10.</td><td colspan="2">deg.</td></tr> +<tr><td>11.</td><td>un ar ddeg</td><td>= 1 + 10.</td></tr> +<tr><td>12.</td><td>deuddeg</td><td>= 2 + 10.</td></tr> +<tr><td>13.</td><td>tri ar ddeg</td><td>= 3 + 10.</td></tr> +<tr><td>14.</td><td>pedwar ar ddeg</td><td>= 4 + 10.</td></tr> +<tr><td>15.</td><td>pymtheg</td><td>= 5 + 10.</td></tr> +<tr><td>16.</td><td>un ar bymtheg</td><td>= 1 + 5 + 10.</td></tr> +<tr><td>17.</td><td>dau ar bymtheg</td><td>= 2 + 5 + 10.</td></tr> +<tr><td>18.</td><td>tri ar bymtheg</td><td>= 3 + 5 + 10.</td></tr> +<tr><td>19.</td><td>pedwar ar bymtheg</td><td>= 4 + 5 + 10.</td></tr> +<tr><td>20.</td><td colspan="2">ugain.</td></tr> +</table> + +<table class="scale"> +<caption>Nahuatl.<a id="FNA-241" href="#FN-241"><sup>241</sup></a></caption> +<tr><td>1.</td><td colspan="2">ce.</td></tr> +<tr><td>2.</td><td colspan="2">ome.</td></tr> +<tr><td>3.</td><td colspan="2">yei.</td></tr> +<tr><td>4.</td><td colspan="2">naui.</td></tr> +<tr><td>5.</td><td colspan="2">macuilli.</td></tr> +<tr><td>6.</td><td>chiquacen</td><td>= [5] + 1.</td></tr> +<tr><td>7.</td><td>chicome</td><td>= [5] + 2.</td></tr> +<tr><td>8.</td><td>chicuey</td><td>= [5] + 3.</td></tr> +<tr><td>9.</td><td>chiucnaui</td><td>= [5] + 4.</td></tr> +<tr><td>10.</td><td colspan="2">matlactli.</td></tr> +<tr><td>11.</td><td>matlactli oce</td><td>= 10 + 1.</td></tr> +<tr><td>12.</td><td>matlactli omome</td><td>= 10 + 2.</td></tr> +<tr><td>13.</td><td>matlactli omey</td><td>= 10 + 3.</td></tr> +<tr><td>14.</td><td>matlactli onnaui</td><td>= 10 + 4.</td></tr> +<tr><td>15.</td><td colspan="2">caxtolli.</td></tr> +<tr><td>16.</td><td>caxtolli oce</td><td>= 15 + 1.</td></tr> +<tr><td>17.</td><td>caxtolli omome</td><td>= 15 + 2.</td></tr> +<tr><td>18.</td><td>caxtolli omey</td><td>= 15 + 3.</td></tr> +<tr><td>19.</td><td>caxtolli onnaui</td><td>= 15 + 4.</td></tr> +<tr><td>20.</td><td>cempualli</td><td>= 1 account.</td></tr> +</table> + + +<table class="scale"> +<caption><a class="pgnum" id="page-142" title="- 142 -"></a>Canaque<a id="FNA-242" href="#FN-242"><sup>242</sup></a> New Caledonia.</caption> + +<tr><td>1.</td><td colspan="2">chaguin.</td></tr> +<tr><td>2.</td><td colspan="2">carou.</td></tr> +<tr><td>3.</td><td colspan="2">careri.</td></tr> +<tr><td>4.</td><td colspan="2">caboue</td></tr> +<tr><td>5.</td><td colspan="2">cani.</td></tr> +<tr><td>6.</td><td>cani-mon-chaguin</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>cani-mon-carou</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>cani-mon-careri</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>cani-mon-caboue</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td colspan="2">panrere.</td></tr> +<tr><td>11.</td><td>panrere-mon-chaguin</td><td>= 10 + 1.</td></tr> +<tr><td>12.</td><td>panrere-mon-carou</td><td>= 10 + 2.</td></tr> +<tr><td>13.</td><td>panrere-mon-careri</td><td>= 10 + 3.</td></tr> +<tr><td>14.</td><td>panrere-mon-caboue</td><td>= 10 + 4.</td></tr> +<tr><td>15.</td><td>panrere-mon-cani</td><td>= 10 + 5.</td></tr> +<tr><td>16.</td><td>panrere-mon-cani-mon-chaguin</td><td>= 10 + 5 + 1.</td></tr> +<tr><td>17.</td><td>panrere-mon-cani-mon-carou</td><td>= 10 + 5 + 2.</td></tr> +<tr><td>18.</td><td>panrere-mon-cani-mon-careri</td><td>= 10 + 5 + 3.</td></tr> +<tr><td>19.</td><td>panrere-mon-cani-mon-caboue</td><td>= 10 + 5 + 4.</td></tr> +<tr><td>20.</td><td>jaquemo</td><td>= 1 person.</td></tr> +</table> + +<table class="scale"> +<caption>Guato.<a id="FNA-243" href="#FN-243"><sup>243</sup></a></caption> +<tr><td>1.</td><td colspan="2">cenai.</td></tr> +<tr><td>2.</td><td colspan="2">dououni.</td></tr> +<tr><td>3.</td><td colspan="2">coum.</td></tr> +<tr><td>4.</td><td colspan="2">dekai.</td></tr> +<tr><td>5.</td><td colspan="2">quinoui.</td></tr> +<tr><td>6.</td><td>cenai-caicaira</td><td>= 1 on the other?</td></tr> +<tr><td>7.</td><td>dououni-caicaira</td><td>= 2 on the other?</td></tr> +<tr><td>8.</td><td>coum-caicaira</td><td>= 3 on the other?</td></tr> +<tr><td>9.</td><td>dekai-caicaira</td><td>= 4 on the other?</td></tr> +<tr><td>10.</td><td>quinoi-da</td><td>= 5 × 2.</td></tr> +<tr><td>11.</td><td>cenai-ai-caibo</td><td>= 1 + (the) hands.</td></tr> +<tr><td>12.</td><td>dououni-ai-caibo</td><td>= 2 + 10.</td></tr> +<tr><td>13.</td><td>coum-ai-caibo</td><td>= 3 + 10.</td></tr> +<tr><td>14.</td><td>dekai-ai-caibo</td><td>= 4 + 10.</td></tr> +<tr><td>15.</td><td>quin-oibo</td><td>= 5 × 3.</td></tr> +<tr><td>16.</td><td>cenai-ai-quacoibo</td><td>= 1 + 15.</td></tr> +<tr><td>17.</td><td>dououni-ai-quacoibo</td><td>= 2 + 15.</td></tr> +<tr><td>18.</td><td>coum-ai-quacoibo</td><td>= 3 + 15.</td></tr> +<tr><td>19.</td><td>dekai-ai-quacoibo</td><td>= 4 + 15.</td></tr> +<tr><td>20.</td><td>quinoui-ai-quacoibo</td><td>= 5 + 15.</td></tr> +</table> + +<p>The meanings assigned to the numerals 6 to 9 are entirely +conjectural. They obviously mean 1, 2, 3, 4, taken +a second time, and as the meanings I have given are +<a class="pgnum" id="page-143" title="- 143 -"></a>often found in primitive systems, they have, at a venture, +been given here.</p> + +<table class="scale"> +<caption>Lifu, Loyalty Islands.<a id="FNA-244" href="#FN-244"><sup>244</sup></a></caption> +<tr><td>1.</td><td colspan="2">ca.</td></tr> +<tr><td>2.</td><td colspan="2">lue.</td></tr> +<tr><td>3.</td><td colspan="2">koeni.</td></tr> +<tr><td>4.</td><td colspan="2">eke.</td></tr> +<tr><td>5.</td><td colspan="2">tji pi.</td></tr> +<tr><td>6.</td><td>ca ngemen</td><td>= 1 above.</td></tr> +<tr><td>7.</td><td>lue ngemen</td><td>= 2 above.</td></tr> +<tr><td>8.</td><td>koeni ngemen</td><td>= 3 above.</td></tr> +<tr><td>9.</td><td>eke ngemen</td><td>= 4 above.</td></tr> +<tr><td>10.</td><td>lue pi</td><td>= 2 × 5.</td></tr> +<tr><td>11.</td><td colspan="2">ca ko.</td></tr> +<tr><td>12.</td><td colspan="2">lue ko.</td></tr> +<tr><td>13.</td><td colspan="2">koeni ko.</td></tr> +<tr><td>14.</td><td colspan="2">eke ko.</td></tr> +<tr><td>15.</td><td>koeni pi</td><td>= 3 × 5.</td></tr> +<tr><td>16.</td><td colspan="2">ca huai ano.</td></tr> +<tr><td>17.</td><td colspan="2">lua huai ano.</td></tr> +<tr><td>18.</td><td colspan="2">koeni huai ano.</td></tr> +<tr><td>19.</td><td colspan="2">eke huai ano.</td></tr> +<tr><td>20.</td><td>ca atj</td><td>= 1 man.</td></tr> +</table> + +<table class="scale"> +<caption>Bongo.<a id="FNA-245" href="#FN-245"><sup>245</sup></a></caption> +<tr><td>1.</td><td colspan="2">kotu.</td></tr> +<tr><td>2.</td><td colspan="2">ngorr.</td></tr> +<tr><td>3.</td><td colspan="2">motta.</td></tr> +<tr><td>4.</td><td colspan="2">neheo.</td></tr> +<tr><td>5.</td><td colspan="2">mui.</td></tr> +<tr><td>6.</td><td>dokotu</td><td>= [5] + 1.</td></tr> +<tr><td>7.</td><td>dongorr</td><td>= [5] + 2.</td></tr> +<tr><td>8.</td><td>domotta</td><td>= [5] + 3.</td></tr> +<tr><td>9.</td><td>doheo</td><td>= [5] + 4.</td></tr> +<tr><td>10.</td><td colspan="2">kih.</td></tr> +<tr><td>11.</td><td>ki dokpo kotu</td><td>= 10 + 1.</td></tr> +<tr><td>12.</td><td>ki dokpo ngorr</td><td>= 10 + 2.</td></tr> +<tr><td>13.</td><td>ki dokpo motta</td><td>= 10 + 3.</td></tr> +<tr><td>14.</td><td>ki dokpo neheo</td><td>= 10 + 4.</td></tr> +<tr><td>15.</td><td>ki dokpo mui</td><td>= 10 + 5.</td></tr> +<tr><td>16.</td><td>ki dokpo mui do mui okpo kotu</td><td>= 10 + 5 more, to 5, 1 more.</td></tr> +<tr><td>17.</td><td>ki dokpo mui do mui okpo ngorr</td><td>= 10 + 5 more, to 5, 2 more.</td></tr> +<tr><td>18.</td><td>ki dokpo mui do mui okpo motta</td><td>= 10 + 5 more, to 5, 3 more.</td></tr> +<tr><td>19.</td><td>ki dokpo mui do mui okpo nehea</td><td>= 10 + 5 more, to 5, 4 more.</td></tr> +<tr><td>20.</td><td colspan="2">mbaba kotu.</td></tr> +</table> + +<p>Above 20, the Lufu and the Bongo systems are vigesimal, +so that they are, as a whole, mixed systems.</p> + +<p><a class="pgnum" id="page-144" title="- 144 -"></a>The Welsh scale begins as though it were to present +a pure decimal structure, and no hint of the quinary +element appears until it has passed 15. The Nahuatl, +on the other hand, counts from 5 to 10 by the ordinary +quinary method, and then appears to pass into the decimal +form. But when 16 is reached, we find the quinary +influence still persistent; and from this point to 20, the +numeral words in both scales are such as to show that +the notion of counting by fives is quite as prominent as +the notion of referring to 10 as a base. Above 20 the +systems become vigesimal, with a quinary or decimal +structure appearing in all numerals except multiples of +20. Thus, in Welsh, 36 is <i>unarbymtheg ar ugain</i>, 1 + 5 + + 10 + 20; and in Nahuatl the same number is <i>cempualli +caxtolli oce</i>, 20 + 15 + 1. Hence these and similar number +systems, though commonly alluded to as vigesimal, +are really mixed scales, with 20 as their primary base. +The Canaque scale differs from the Nahuatl only in +forming a compound word for 15, instead of introducing +a new and simple term.</p> + +<p>In the examples which follow, it is not thought best +to extend the lists of numerals beyond 10, except in +special instances where the illustration of some particular +point may demand it. The usual quinary scale will be +found, with a few exceptions like those just instanced, +to have the following structure or one similar to it in all +essential details: 1, 2, 3, 4, 5, 5-1, 5-2, 5-3, 5-4, 10, +10-1, 10-2, 10-3, 10-4, 10-5, 10-5-1, 10-5-2, 10-5-3, +10-5-4, 20. From these forms the entire system can +<a class="pgnum" id="page-145" title="- 145 -"></a>readily be constructed as soon as it is known whether +its principal base is to be 10 or 20.</p> + +<p>Turning first to the native African languages, I have +selected the following quinary scales from the abundant +material that has been collected by the various explorers +of the “Dark Continent.” In some cases the numerals +of certain tribes, as given by one writer, are found to +differ widely from the same numerals as reported by +another. No attempt has been made at comparison of +these varying forms of orthography, which are usually +to be ascribed to difference of nationality on the part +of the collectors.</p> + +<table class="scale"> +<caption>Feloops.<a id="FNA-246" href="#FN-246"><sup>246</sup></a></caption> +<tr><td>1.</td><td colspan="2">enory.</td></tr> +<tr><td>2.</td><td colspan="2">sickaba, or cookaba.</td></tr> +<tr><td>3.</td><td colspan="2">sisajee.</td></tr> +<tr><td>4.</td><td colspan="2">sibakeer.</td></tr> +<tr><td>5.</td><td colspan="2">footuck.</td></tr> +<tr><td>6.</td><td>footuck-enory</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>footuck-cookaba</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>footuck-sisajee</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>footuck-sibakeer</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">sibankonyen.</td></tr> +</table> + +<table class="scale"> +<caption>Kissi.<a id="FNA-247" href="#FN-247"><sup>247</sup></a></caption> +<tr><td>1.</td><td colspan="2">pili.</td></tr> +<tr><td>2.</td><td colspan="2">miu.</td></tr> +<tr><td>3.</td><td colspan="2">nga.</td></tr> +<tr><td>4.</td><td colspan="2">iol.</td></tr> +<tr><td>5.</td><td colspan="2">nguenu.</td></tr> +<tr><td>6.</td><td>ngom-pum</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>ngom-miu</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>ngommag</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>nguenu-iol</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">to.</td></tr> +</table> + +<table class="scale"> +<caption>Ashantee.<a id="FNA-248" href="#FN-248"><sup>248</sup></a></caption> +<tr><td>1.</td><td colspan="2">tah.</td></tr> +<tr><td>2.</td><td colspan="2">noo.</td></tr> +<tr><td>3.</td><td colspan="2">sah.</td></tr> +<tr><td>4.</td><td colspan="2">nah.</td></tr> +<tr><td>5.</td><td colspan="2">taw.</td></tr> +<tr><td>6.</td><td>torata</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>toorifeenoo</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>toorifeessa</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>toorifeena</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td colspan="2">nopnoo.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-146" title="- 146 -"></a>Basa.<a id="FNA-249" href="#FN-249"><sup>249</sup></a></caption> +<tr><td>1.</td><td colspan="2">do.</td></tr> +<tr><td>2.</td><td colspan="2">so.</td></tr> +<tr><td>3.</td><td colspan="2">ta.</td></tr> +<tr><td>4.</td><td colspan="2">hinye.</td></tr> +<tr><td>5.</td><td colspan="2">hum.</td></tr> +<tr><td>6.</td><td>hum-le-do</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>hum-le-so</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>hum-le-ta</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>hum-le-hinyo</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td colspan="2">bla-bue.</td></tr> +</table> + +<table class="scale"> +<caption>Jallonkas.<a id="FNA-250" href="#FN-250"><sup>250</sup></a></caption> +<tr><td>1.</td><td colspan="2">kidding.</td></tr> +<tr><td>2.</td><td colspan="2">fidding.</td></tr> +<tr><td>3.</td><td colspan="2">sarra.</td></tr> +<tr><td>4.</td><td colspan="2">nani.</td></tr> +<tr><td>5.</td><td colspan="2">soolo.</td></tr> +<tr><td>6.</td><td colspan="2">seni.</td></tr> +<tr><td>7.</td><td>soolo ma fidding</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>soolo ma sarra</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>soolo ma nani</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td colspan="2">nuff.</td></tr> +</table> + +<table class="scale"> +<caption>Kru.</caption> +<tr><td>1.</td><td colspan="2">da-do.</td></tr> +<tr><td>2.</td><td colspan="2">de-son.</td></tr> +<tr><td>3.</td><td colspan="2">de-tan.</td></tr> +<tr><td>4.</td><td colspan="2">de-nie.</td></tr> +<tr><td>5.</td><td colspan="2">de-mu.</td></tr> +<tr><td>6.</td><td>dme-du</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>ne-son</td><td>= [5] + 2.</td></tr> +<tr><td>8.</td><td>ne-tan</td><td>= [5] + 3.</td></tr> +<tr><td>9.</td><td>sepadu</td><td>= 10 − 1?</td></tr> +<tr><td>10.</td><td colspan="2">pua.</td></tr> +</table> + +<table class="scale"> +<caption>Jaloffs.<a id="FNA-251" href="#FN-251"><sup>251</sup></a></caption> +<tr><td>1.</td><td colspan="2">wean.</td></tr> +<tr><td>2.</td><td colspan="2">yar.</td></tr> +<tr><td>3.</td><td colspan="2">yat.</td></tr> +<tr><td>4.</td><td colspan="2">yanet.</td></tr> +<tr><td>5.</td><td colspan="2">judom.</td></tr> +<tr><td>6.</td><td>judom-wean</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>judom-yar</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>judom-yat</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>judom yanet</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">fook.</td></tr> +</table> + + +<table class="scale"> +<caption>Golo.<a id="FNA-252" href="#FN-252"><sup>252</sup></a></caption> +<tr><td>1.</td><td colspan="2">mbali.</td></tr> +<tr><td>2.</td><td colspan="2">bisi.</td></tr> +<tr><td>3.</td><td colspan="2">bitta.</td></tr> +<tr><td>4.</td><td colspan="2">banda.</td></tr> +<tr><td>5.</td><td colspan="2">zonno.</td></tr> +<tr><td>6.</td><td>tsimmi tongbali</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>tsimmi tobisi</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>tsimmi tobitta</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>tsimmi to banda</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td colspan="2">nifo.</td></tr> + +</table> + + +<table class="scale"> +<caption><a class="pgnum" id="page-147" title="- 147 -"></a>Foulah.<a id="FNA-253" href="#FN-253"><sup>253</sup></a></caption> + +<tr><td>1.</td><td colspan="2">go.</td></tr> +<tr><td>2.</td><td colspan="2">deeddee.</td></tr> +<tr><td>3.</td><td colspan="2">tettee.</td></tr> +<tr><td>4.</td><td colspan="2">nee.</td></tr> +<tr><td>5.</td><td colspan="2">jouee.</td></tr> +<tr><td>6.</td><td>jego</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>jedeeddee</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>je-tettee</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>je-nee</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">sappo.</td></tr> +</table> + +<table class="scale"> +<caption>Soussou.<a id="FNA-254" href="#FN-254"><sup>254</sup></a></caption> +<tr><td>1.</td><td colspan="2">keren.</td></tr> +<tr><td>2.</td><td colspan="2">firing.</td></tr> +<tr><td>3.</td><td colspan="2">sarkan.</td></tr> +<tr><td>4.</td><td colspan="2">nani.</td></tr> +<tr><td>5.</td><td colspan="2">souli.</td></tr> +<tr><td>6.</td><td colspan="2">seni.</td></tr> +<tr><td>7.</td><td>solo-fere</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>solo-mazarkan</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>solo-manani</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td colspan="2">fu.</td></tr> +</table> + +<table class="scale"> +<caption>Bullom.<a id="FNA-255" href="#FN-255"><sup>255</sup></a></caption> +<tr><td>1.</td><td colspan="2">bul.</td></tr> +<tr><td>2.</td><td colspan="2">tin.</td></tr> +<tr><td>3.</td><td colspan="2">ra.</td></tr> +<tr><td>4.</td><td colspan="2">hyul.</td></tr> +<tr><td>5.</td><td colspan="2">men.</td></tr> +<tr><td>6.</td><td>men-bul</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>men-tin</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>men-ra</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>men-hyul</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">won.</td></tr> +</table> + +<table class="scale"> +<caption>Vei.<a id="FNA-256" href="#FN-256"><sup>256</sup></a></caption> +<tr><td>1.</td><td colspan="2">dondo.</td></tr> +<tr><td>2.</td><td colspan="2">fera.</td></tr> +<tr><td>3.</td><td colspan="2">sagba.</td></tr> +<tr><td>4.</td><td colspan="2">nani.</td></tr> +<tr><td>5.</td><td colspan="2">soru.</td></tr> +<tr><td>6.</td><td>sun-dondo</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>sum-fera</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>sun-sagba</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>sun-nani</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">tan.</td></tr> +</table> + +<table class="scale"> +<caption>Dinka.<a id="FNA-257" href="#FN-257"><sup>257</sup></a></caption> +<tr><td>1.</td><td colspan="2">tok.</td></tr> +<tr><td>2.</td><td colspan="2">rou.</td></tr> +<tr><td>3.</td><td colspan="2">dyak.</td></tr> +<tr><td>4.</td><td colspan="2">nuan.</td></tr> +<tr><td>5.</td><td colspan="2">wdyets.</td></tr> +<tr><td>6.</td><td>wdetem</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>wderou</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>bet, bed</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>wdenuan</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td>wtyer</td><td>= 5 × 2.</td></tr> +</table> + + + + +<table class="scale"> +<caption><a class="pgnum" id="page-148" title="- 148 -"></a>Temne.</caption> +<tr><td>1.</td><td colspan="2">in.</td></tr> +<tr><td>2.</td><td colspan="2">ran.</td></tr> +<tr><td>3.</td><td colspan="2">sas.</td></tr> +<tr><td>4.</td><td colspan="2">anle.</td></tr> +<tr><td>5.</td><td colspan="2">tr-amat.</td></tr> +<tr><td>6.</td><td>tr-amat rok-in</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>tr-amat de ran</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>tr-amat re sas</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>tr-amat ro n-anle</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td colspan="2">tr-ofatr.</td></tr> +</table> + +<table class="scale"> +<caption>Abaker.<a id="FNA-258" href="#FN-258"><sup>258</sup></a></caption> +<tr><td>1.</td><td colspan="2">kili.</td></tr> +<tr><td>2.</td><td colspan="2">bore.</td></tr> +<tr><td>3.</td><td colspan="2">dotla.</td></tr> +<tr><td>4.</td><td colspan="2">ashe.</td></tr> +<tr><td>5.</td><td colspan="2">ini.</td></tr> +<tr><td>6.</td><td>im kili</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>im-bone</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>ini-dotta</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>tin ashe</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">chica.</td></tr> +</table> + + +<table class="scale"> +<caption>Bagrimma.<a id="FNA-259" href="#FN-259"><sup>259</sup></a></caption> +<tr><td>1.</td><td colspan="2">kede.</td></tr> +<tr><td>2.</td><td colspan="2">sab.</td></tr> +<tr><td>3.</td><td colspan="2">muta.</td></tr> +<tr><td>4.</td><td colspan="2">so.</td></tr> +<tr><td>5.</td><td colspan="2">mi.</td></tr> +<tr><td>6.</td><td>mi-ga</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td colspan="2">tsidi.</td></tr> +<tr><td>8.</td><td>marta</td><td>= 5 + <del title="artithmetically wrong">2</del><ins title="replaced 2 with 3 for the sake of arithmetical correctness and in agreement with previous numerals">3</ins>.</td></tr> +<tr><td>9.</td><td>do-so</td><td>= [5] + <del title="arithmetically wrong">3</del><ins title="replaced 3 with 4 for the sake of arithmetical correctness and in agreement with previous numerals">4</ins></td></tr> +<tr><td>10.</td><td colspan="2">duk-keme.</td></tr> +</table> + +<table class="scale"> +<caption>Papaa.<a id="FNA-260" href="#FN-260"><sup>260</sup></a></caption> +<tr><td>1.</td><td colspan="2">depoo.</td></tr> +<tr><td>2.</td><td colspan="2">auwi.</td></tr> +<tr><td>3.</td><td colspan="2">ottong.</td></tr> +<tr><td>4.</td><td colspan="2">enne.</td></tr> +<tr><td>5.</td><td colspan="2">attong.</td></tr> +<tr><td>6.</td><td colspan="2">attugo.</td></tr> +<tr><td>7.</td><td>atjuwe</td><td>= [5] + 2.</td></tr> +<tr><td>8.</td><td>attiatong</td><td>= [5] + 3.</td></tr> +<tr><td>9.</td><td>atjeenne</td><td>= [5] + 4.</td></tr> +<tr><td>10.</td><td colspan="2">awo.</td></tr> +</table> + + +<table class="scale"> +<caption>Efik.<a id="FNA-261" href="#FN-261"><sup>261</sup></a></caption> +<tr><td>1.</td><td colspan="2">kiet.</td></tr> +<tr><td>2.</td><td colspan="2">iba.</td></tr> +<tr><td>3.</td><td colspan="2">ita.</td></tr> +<tr><td>4.</td><td colspan="2">inan.</td></tr> +<tr><td>5.</td><td colspan="2">itiun.</td></tr> +<tr><td>6.</td><td>itio-kiet</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>itia-ba</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>itia-eta</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>osu-kiet</td><td>= 10 − 1?</td></tr> +<tr><td>10.</td><td colspan="2">duup.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-149" title="- 149 -"></a>Nupe.<a id="FNA-262" href="#FN-262"><sup>262</sup></a></caption> + +<tr><td>1.</td><td colspan="2">nini.</td></tr> +<tr><td>2.</td><td colspan="2">gu-ba.</td></tr> +<tr><td>3.</td><td colspan="2">gu-ta.</td></tr> +<tr><td>4.</td><td colspan="2">gu-ni.</td></tr> +<tr><td>5.</td><td colspan="2">gu-tsun.</td></tr> +<tr><td>6.</td><td>gu-sua-yin</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>gu-tua-ba</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>gu-tu-ta</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>gu-tua-ni</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td colspan="2">gu-wo.</td></tr> +</table> + +<table class="scale"> +<caption>Mokko.<a id="FNA-263" href="#FN-263"><sup>263</sup></a></caption> + +<tr><td>1.</td><td colspan="2">kiä.</td></tr> +<tr><td>2.</td><td colspan="2">iba.</td></tr> +<tr><td>3.</td><td colspan="2">itta.</td></tr> +<tr><td>4.</td><td colspan="2">inan.</td></tr> +<tr><td>5.</td><td colspan="2">üttin.</td></tr> +<tr><td>6.</td><td>itjüekee</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>ittiaba</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>itteiata</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td colspan="2">huschukiet.</td></tr> +<tr><td>10.</td><td colspan="2">büb.</td></tr> +</table> + +<table class="scale"> +<caption>Kanuri.<a id="FNA-264" href="#FN-264"><sup>264</sup></a></caption> + +<tr><td>1.</td><td colspan="2">tilo.</td></tr> +<tr><td>2.</td><td colspan="2">ndi.</td></tr> +<tr><td>3.</td><td colspan="2">yasge.</td></tr> +<tr><td>4.</td><td colspan="2">dege.</td></tr> +<tr><td>5.</td><td colspan="2">ugu.</td></tr> +<tr><td>6.</td><td>arasge</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td colspan="2">tulur.</td></tr> +<tr><td>8.</td><td>wusge</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td colspan="2">legar.</td></tr> +<tr><td>10.</td><td>megu</td><td>= 2 × 5.</td></tr> +</table> + +<table class="scale"> +<caption>Binin.<a id="FNA-265" href="#FN-265"><sup>265</sup></a></caption> + +<tr><td>1.</td><td colspan="2">bo.</td></tr> +<tr><td>2.</td><td colspan="2">be.</td></tr> +<tr><td>3.</td><td colspan="2">la.</td></tr> +<tr><td>4.</td><td colspan="2">nin.</td></tr> +<tr><td>5.</td><td colspan="2">tang.</td></tr> +<tr><td>6.</td><td>tahu</td><td>= 5 + 1?</td></tr> +<tr><td>7.</td><td>tabi</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>tara</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>ianin (tanin?)</td><td>= 5 + 4?</td></tr> +<tr><td>10.</td><td colspan="2">te.</td></tr> +</table> + +<table class="scale"> +<caption>Kredy.<a id="FNA-266" href="#FN-266"><sup>266</sup></a></caption> + +<tr><td>1.</td><td colspan="2">baia.</td></tr> +<tr><td>2.</td><td colspan="2">rommu.</td></tr> +<tr><td>3.</td><td colspan="2">totto.</td></tr> +<tr><td>4.</td><td colspan="2">sosso.</td></tr> +<tr><td>5.</td><td colspan="2">saya.</td></tr> +<tr><td>6.</td><td>yembobaia</td><td>= [5] + 1.</td></tr> +<tr><td>7.</td><td>yemborommu</td><td>= [5] + 2.</td></tr> +<tr><td>8.</td><td>yembototto</td><td>= [5] + 3.</td></tr> +<tr><td>9.</td><td>yembososso</td><td>= [5] + 4.</td></tr> +<tr><td>10.</td><td colspan="2">puh.</td></tr> +</table> + + +<table class="scale"> +<caption><a class="pgnum" id="page-150" title="- 150 -"></a>Herero.<a id="FNA-267" href="#FN-267"><sup>267</sup></a></caption> + +<tr><td>1.</td><td colspan="2">mue.</td></tr> +<tr><td>2.</td><td colspan="2">vari.</td></tr> +<tr><td>3.</td><td colspan="2">tatu.</td></tr> +<tr><td>4.</td><td colspan="2">ne.</td></tr> +<tr><td>5.</td><td colspan="2">tano.</td></tr> +<tr><td>6.</td><td>hambou-mue</td><td>= [5] + 1.</td></tr> +<tr><td>7.</td><td>hambou-vari</td><td>= [5] + 2.</td></tr> +<tr><td>8.</td><td>hambou-tatu</td><td>= [5] + 3.</td></tr> +<tr><td>9.</td><td>hambou-ne</td><td>= [5] + 4.</td></tr> +<tr><td>10.</td><td colspan="2"></td></tr> +</table> + +<table class="scale"> +<caption>Ki-Yau.<a id="FNA-268" href="#FN-268"><sup>268</sup></a></caption> + +<tr><td>1.</td><td colspan="2">jumo.</td></tr> +<tr><td>2.</td><td colspan="2">wawiri.</td></tr> +<tr><td>3.</td><td colspan="2">watatu.</td></tr> +<tr><td>4.</td><td colspan="2">mcheche.</td></tr> +<tr><td>5.</td><td colspan="2">msano.</td></tr> +<tr><td>6.</td><td>musano na jumo</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>musano na wiri</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>musano na watatu</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>musano na mcheche</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td colspan="2">ikumi.</td></tr> +</table> + +<table class="scale"> +<caption>Fernando Po.<a id="FNA-269" href="#FN-269"><sup>269</sup></a></caption> + +<tr><td>1.</td><td colspan="2">muli.</td></tr> +<tr><td>2.</td><td colspan="2">mempa.</td></tr> +<tr><td>3.</td><td colspan="2">meta.</td></tr> +<tr><td>4.</td><td colspan="2">miene.</td></tr> +<tr><td>5.</td><td colspan="2">mimito.</td></tr> +<tr><td>6.</td><td>mimito na muli</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>mimito na mempa</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>mimito na meta</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>mimito na miene</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td>miemieu</td><td>= 5-5?</td></tr> +</table> + +<table class="scale"> +<caption>Ki-Nyassa</caption> + +<tr><td>1.</td><td colspan="2">kimodzi.</td></tr> +<tr><td>2.</td><td colspan="2">vi-wiri.</td></tr> +<tr><td>3.</td><td colspan="2">vi-tatu.</td></tr> +<tr><td>4.</td><td colspan="2">vinye.</td></tr> +<tr><td>5.</td><td colspan="2">visano.</td></tr> +<tr><td>6.</td><td>visano na kimodzi</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>visano na vi-wiri</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>visano na vitatu</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>visano na vinye</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td colspan="2">chikumi.</td></tr> +</table> + +<table class="scale"> +<caption>Balengue.<a id="FNA-270" href="#FN-270"><sup>270</sup></a></caption> + +<tr><td>1.</td><td colspan="2">guevoho.</td></tr> +<tr><td>2.</td><td colspan="2">ibare.</td></tr> +<tr><td>3.</td><td colspan="2">raro.</td></tr> +<tr><td>4.</td><td colspan="2">inaï.</td></tr> +<tr><td>5.</td><td colspan="2">itano.</td></tr> +<tr><td>6.</td><td>itano na guevoho</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>itano na ibare</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>itano na raro</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>itano na inaï</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td colspan="2">ndioum, or nai-hinaï.</td></tr> +</table> + + +<table class="scale"> +<caption><a class="pgnum" id="page-151" title="- 151 -"></a>Kunama.<a id="FNA-271" href="#FN-271"><sup>271</sup></a></caption> + +<tr><td>1.</td><td colspan="2">ella.</td></tr> +<tr><td>2.</td><td colspan="2">bare.</td></tr> +<tr><td>3.</td><td colspan="2">sadde.</td></tr> +<tr><td>4.</td><td colspan="2">salle.</td></tr> +<tr><td>5.</td><td colspan="2">kussume.</td></tr> +<tr><td>6.</td><td>kon-t'-ella</td><td>= hand 1.</td></tr> +<tr><td>7.</td><td>kon-te-bare</td><td>= hand 2.</td></tr> +<tr><td>8.</td><td>kon-te-sadde</td><td>= hand 3.</td></tr> +<tr><td>9.</td><td>kon-te-salle</td><td>= hand 4.</td></tr> +<tr><td>10.</td><td colspan="2">kol-lakada.</td></tr> +</table> + + +<table class="scale"> +<caption>Gola.<a id="FNA-272" href="#FN-272"><sup>272</sup></a></caption> + +<tr><td>1.</td><td colspan="2">ngoumou.</td></tr> +<tr><td>2.</td><td colspan="2">ntie.</td></tr> +<tr><td>3.</td><td colspan="2">ntaï.</td></tr> +<tr><td>4.</td><td colspan="2">tina.</td></tr> +<tr><td>5.</td><td colspan="2">nonon.</td></tr> +<tr><td>6.</td><td>diegoum</td><td>= [5] + 1.</td></tr> +<tr><td>7.</td><td>dientie</td><td>= [5] + 2.</td></tr> +<tr><td>8.</td><td>dietai</td><td>= [5] + 3.</td></tr> +<tr><td>9.</td><td>dectina</td><td>= [5] + 4.</td></tr> +<tr><td>10.</td><td colspan="2">esia.</td></tr> +</table> + + +<table class="scale"> +<caption>Barea.<a id="FNA-273" href="#FN-273"><sup>273</sup></a></caption> + +<tr><td>1.</td><td colspan="2">doko</td></tr> +<tr><td>2.</td><td colspan="2">arega.</td></tr> +<tr><td>3.</td><td colspan="2">sane.</td></tr> +<tr><td>4.</td><td colspan="2">sone.</td></tr> +<tr><td>5.</td><td colspan="2">oita.</td></tr> +<tr><td>6.</td><td colspan="2">data.</td></tr> +<tr><td>7.</td><td>dz-ariga</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>dis-sena</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>lefete-mada</td><td>= without 10.</td></tr> +<tr><td>10.</td><td colspan="2">lefek.</td></tr> +</table> + + +<table class="scale"> +<caption>Matibani.<a id="FNA-274" href="#FN-274"><sup>274</sup></a></caption> + +<tr><td>1.</td><td colspan="2">mosa.</td></tr> +<tr><td>2.</td><td colspan="2">pili.</td></tr> +<tr><td>3.</td><td colspan="2">taru.</td></tr> +<tr><td>4.</td><td colspan="2">teje.</td></tr> +<tr><td>5.</td><td colspan="2">taru.</td></tr> +<tr><td>6.</td><td>tana mosa</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>tana pili</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>tana taru</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td colspan="2">loco.</td></tr> +<tr><td>10.</td><td colspan="2">loco nakege.</td></tr> +</table> + + +<table class="scale"> +<caption>Bonzé.<a id="FNA-275" href="#FN-275"><sup>275</sup></a></caption> + +<tr><td>1.</td><td colspan="2">tan.</td></tr> +<tr><td>2.</td><td colspan="2">vele.</td></tr> +<tr><td>3.</td><td colspan="2">daba.</td></tr> +<tr><td>4.</td><td colspan="2">nani.</td></tr> +<tr><td>5.</td><td colspan="2">lolou.</td></tr> +<tr><td>6.</td><td>maïda</td><td>= [5] + 1.</td></tr> +<tr><td>7.</td><td>maïfile</td><td>= [5] + 2.</td></tr> +<tr><td>8.</td><td>maïshaba</td><td>= [5] + 3.</td></tr> +<tr><td>9.</td><td>maïnan</td><td>= [5] + 4.</td></tr> +<tr><td>10.</td><td colspan="2">bou.</td></tr> +</table> + + +<table class="scale"> +<caption><a class="pgnum" id="page-152" title="- 152 -"></a>Mpovi</caption> + +<tr><td>1.</td><td colspan="2">moueta.</td></tr> +<tr><td>2.</td><td colspan="2">bevali.</td></tr> +<tr><td>3.</td><td colspan="2">betata.</td></tr> +<tr><td>4.</td><td colspan="2">benaï.</td></tr> +<tr><td>5.</td><td colspan="2">betani.</td></tr> +<tr><td>6.</td><td>betani moueta</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>betani bevali</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>betani betata</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>betani benai</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">nchinia.</td></tr> +</table> + +<table class="scale"> +<caption>Triton's Bay, New Quinea.<a id="FNA-276" href="#FN-276"><sup>276</sup></a></caption> + +<tr><td>1.</td><td colspan="2">samosi.</td></tr> +<tr><td>2.</td><td colspan="2">roueti.</td></tr> +<tr><td>3.</td><td colspan="2">tourou.</td></tr> +<tr><td>4.</td><td colspan="2">faat.</td></tr> +<tr><td>5.</td><td colspan="2">rimi.</td></tr> +<tr><td>6.</td><td>rim-samosi</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>rim-roueti</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>rim-tourou</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>rim-faat</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">outsia.</td></tr> +</table> + +<table class="scale"> +<caption>Ende, or Flores.<a id="FNA-277" href="#FN-277"><sup>277</sup></a></caption> + +<tr><td>1.</td><td colspan="2">sa.</td></tr> +<tr><td>2.</td><td colspan="2">zua.</td></tr> +<tr><td>3.</td><td colspan="2">telu.</td></tr> +<tr><td>4.</td><td colspan="2">wutu.</td></tr> +<tr><td>5.</td><td>lima</td><td>= hand.</td></tr> +<tr><td>6.</td><td>lima-sa</td><td>= 5-1, or hand 1.</td></tr> +<tr><td>7.</td><td>lima-zua</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>rua-butu</td><td>= 2 × 4?</td></tr> +<tr><td>9.</td><td>trasa</td><td>= [10] − 1?</td></tr> +<tr><td>10.</td><td colspan="2">sabulu.</td></tr> +</table> + + +<table class="scale"> +<caption>Mallicolo.<a id="FNA-278" href="#FN-278"><sup>278</sup></a></caption> + +<tr><td>1.</td><td colspan="2">tseekaee.</td></tr> +<tr><td>2.</td><td colspan="2">ery.</td></tr> +<tr><td>3.</td><td colspan="2">erei.</td></tr> +<tr><td>4.</td><td colspan="2">ebats.</td></tr> +<tr><td>5.</td><td colspan="2">ereem.</td></tr> +<tr><td>6.</td><td>tsookaee</td><td>= [5] + 1.</td></tr> +<tr><td>7.</td><td>gooy</td><td>= [5] + 2.</td></tr> +<tr><td>8.</td><td>hoorey</td><td>= [5] + 3.</td></tr> +<tr><td>9.</td><td>goodbats</td><td>= [5] + 4.</td></tr> +<tr><td>10.</td><td colspan="2">senearn.</td></tr> +</table> + + +<table class="scale"> +<caption>Ebon, Marshall Islands.<a id="FNA-279" href="#FN-279"><sup>279</sup></a></caption> + +<tr><td>1.</td><td colspan="2">iuwun.</td></tr> +<tr><td>2.</td><td colspan="2">drud.</td></tr> +<tr><td>3.</td><td colspan="2">chilu.</td></tr> +<tr><td>4.</td><td colspan="2">emer.</td></tr> +<tr><td>5.</td><td colspan="2">lailem.</td></tr> +<tr><td>6.</td><td>chilchinu</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>chilchime</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>twalithuk</td><td>= [10] − 2.</td></tr> +<tr><td>9.</td><td>twahmejuwou</td><td>= [10] − 1.</td></tr> +<tr><td>10.</td><td colspan="2">iungou.</td></tr> +</table> + + +<table class="scale"> +<caption><a class="pgnum" id="page-153" title="- 153 -"></a>Uea, Loyalty Island.<a id="FNA-280" href="#FN-280"><sup>280</sup></a></caption> + +<tr><td>1.</td><td>tahi.</td></tr> +<tr><td>2.</td><td>lua.</td></tr> +<tr><td>3.</td><td>tolu.</td></tr> +<tr><td>4.</td><td>fa.</td></tr> +<tr><td>5.</td><td>lima.</td></tr> +<tr><td>6.</td><td>tahi.</td></tr> +<tr><td>7.</td><td>lua.</td></tr> +<tr><td>8.</td><td>tolu.</td></tr> +<tr><td>9.</td><td>fa.</td></tr> +<tr><td>10.</td><td>lima.</td></tr> +</table> + +<table class="scale"> +<caption>Uea.<a href="#FN-280"><sup>280</sup></a><span>—[another dialect.]</span></caption> + +<tr><td>1.</td><td colspan="2">hacha.</td></tr> +<tr><td>2.</td><td colspan="2">lo.</td></tr> +<tr><td>3.</td><td colspan="2">kuun.</td></tr> +<tr><td>4.</td><td colspan="2">thack.</td></tr> +<tr><td>5.</td><td colspan="2">thabumb.</td></tr> +<tr><td>6.</td><td>lo-acha</td><td>= 2d 1.</td></tr> +<tr><td>7.</td><td>lo-alo</td><td>= 2d 2.</td></tr> +<tr><td>8.</td><td>lo-kuun</td><td>= 2d 3.</td></tr> +<tr><td>9.</td><td>lo-thack</td><td>= 2d 4.</td></tr> +<tr><td>10.</td><td colspan="2">lebenetee.</td></tr> +</table> + +<table class="scale"> +<caption>Isle of Pines.<a id="FNA-281" href="#FN-281"><sup>281</sup></a></caption> + +<tr><td>1.</td><td colspan="2">ta.</td></tr> +<tr><td>2.</td><td colspan="2">bo.</td></tr> +<tr><td>3.</td><td colspan="2">beti.</td></tr> +<tr><td>4.</td><td colspan="2">beu.</td></tr> +<tr><td>5.</td><td colspan="2">ta-hue.</td></tr> +<tr><td>6.</td><td>no-ta</td><td>= 2d 1.</td></tr> +<tr><td>7.</td><td>no-bo</td><td>= 2d 2.</td></tr> +<tr><td>8.</td><td>no-beti</td><td>= 2d 3.</td></tr> +<tr><td>9.</td><td>no-beu</td><td>= 2d 4.</td></tr> +<tr><td>10.</td><td colspan="2">de-kau.</td></tr> +</table> + +<table class="scale"> +<caption>Ureparapara, Banks Islands.<a id="FNA-282" href="#FN-282"><sup>282</sup></a></caption> + +<tr><td>1.</td><td colspan="2">vo towa.</td></tr> +<tr><td>2.</td><td colspan="2">vo ro.</td></tr> +<tr><td>3.</td><td colspan="2">vo tol.</td></tr> +<tr><td>4.</td><td colspan="2">vo vet.</td></tr> +<tr><td>5.</td><td>teveliem</td><td>= 1 hand.</td></tr> +<tr><td>6.</td><td>leve jea</td><td>= other 1.</td></tr> +<tr><td>7.</td><td>leve ro</td><td>= other 2.</td></tr> +<tr><td>8.</td><td>leve tol</td><td>= other 3.</td></tr> +<tr><td>9.</td><td>leve vet</td><td>= other 4.</td></tr> +<tr><td>10.</td><td>sanowul</td><td>= 2 sets.</td></tr> +</table> + +<table class="scale"> +<caption>Mota, Banks Islands.<a href="#FN-282"><sup>282</sup></a></caption> + +<tr><td>1.</td><td colspan="2">tuwale.</td></tr> +<tr><td>2.</td><td colspan="2">nirua.</td></tr> +<tr><td>3.</td><td colspan="2">nitol.</td></tr> +<tr><td>4.</td><td colspan="2">nivat.</td></tr> +<tr><td>5.</td><td>tavelima</td><td>= 1 hand.</td></tr> +<tr><td>6.</td><td>laveatea</td><td>= other 1.</td></tr> +<tr><td>7.</td><td>lavearua</td><td>= other 2.</td></tr> +<tr><td>8.</td><td>laveatol</td><td>= other 3.</td></tr> +<tr><td>9.</td><td>laveavat</td><td>= other 4.</td></tr> +<tr><td>10.</td><td>sanavul</td><td>= 2 sets.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-154" title="- 154 -"></a>New Caledonia.<a id="FNA-283" href="#FN-283"><sup>283</sup></a></caption> + +<tr><td>1.</td><td colspan="2">parai.</td></tr> +<tr><td>2.</td><td colspan="2">paroo.</td></tr> +<tr><td>3.</td><td colspan="2">parghen.</td></tr> +<tr><td>4.</td><td colspan="2">parbai.</td></tr> +<tr><td>5.</td><td colspan="2">panim.</td></tr> +<tr><td>6.</td><td>panim-gha</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>panim-roo</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>panim-ghen</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>panim-bai</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">parooneek.</td></tr> +</table> + +<table class="scale"> +<caption>Yengen, New Cal.<a id="FNA-284" href="#FN-284"><sup>284</sup></a></caption> + +<tr><td>1.</td><td colspan="2">hets.</td></tr> +<tr><td>2.</td><td colspan="2">heluk.</td></tr> +<tr><td>3.</td><td colspan="2">heyen.</td></tr> +<tr><td>4.</td><td colspan="2">pobits.</td></tr> +<tr><td>5.</td><td>nim</td><td>= hand.</td></tr> +<tr><td>6.</td><td>nim-wet</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>nim-weluk</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>nim-weyen</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>nim-pobit</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">pain-duk.</td></tr> +</table> + +<table class="scale"> +<caption>Aneiteum.<a id="FNA-285" href="#FN-285"><sup>285</sup></a></caption> + +<tr><td>1.</td><td colspan="2">ethi.</td></tr> +<tr><td>2.</td><td colspan="2">ero.</td></tr> +<tr><td>3.</td><td colspan="2">eseik.</td></tr> +<tr><td>4.</td><td colspan="2">manohwan.</td></tr> +<tr><td>5.</td><td colspan="2">nikman.</td></tr> +<tr><td>6.</td><td>nikman cled et ethi</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>nikman cled et oro</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>nikman cled et eseik</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>nikman cled et manohwan</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td>nikman lep ikman</td><td>= 5 + 5.</td></tr> +</table> + +<table class="scale"> +<caption>Tanna</caption> + +<tr><td>1.</td><td colspan="2">riti.</td></tr> +<tr><td>2.</td><td colspan="2">karu.</td></tr> +<tr><td>3.</td><td colspan="2">kahar.</td></tr> +<tr><td>4.</td><td colspan="2">kefa.</td></tr> +<tr><td>5.</td><td colspan="2">krirum.</td></tr> +<tr><td>6.</td><td>krirum riti</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>krirum karu</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>krirum kahar?</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>krirum kefa?</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">——</td></tr> +</table> + +<table class="scale"> +<caption>Eromanga</caption> + +<tr><td>1.</td><td colspan="2">sai.</td></tr> +<tr><td>2.</td><td colspan="2">duru.</td></tr> +<tr><td>3.</td><td colspan="2">disil.</td></tr> +<tr><td>4.</td><td colspan="2">divat.</td></tr> +<tr><td>5.</td><td>siklim</td><td>= 1 hand.</td></tr> +<tr><td>6.</td><td>misikai</td><td>= other 1?</td></tr> +<tr><td>7.</td><td>siklim naru</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>siklim disil</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>siklim mindivat</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td>narolim</td><td>= 2 hands.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-155" title="- 155 -"></a>Fate, New Heb.<a id="FNA-286" href="#FN-286"><sup>286</sup></a></caption> + +<tr><td>1.</td><td colspan="2">iskei.</td></tr> +<tr><td>2.</td><td colspan="2">rua.</td></tr> +<tr><td>3.</td><td colspan="2">tolu.</td></tr> +<tr><td>4.</td><td colspan="2">bate.</td></tr> +<tr><td>5.</td><td>lima</td><td>= hand.</td></tr> +<tr><td>6.</td><td>la tesa</td><td>= other 1.</td></tr> +<tr><td>7.</td><td>la rua</td><td>= other 2.</td></tr> +<tr><td>8.</td><td>la tolu</td><td>= other 3.</td></tr> +<tr><td>9.</td><td>la fiti</td><td>= other 4.</td></tr> +<tr><td>10.</td><td>relima</td><td>= 2 hands.</td></tr> +</table> + +<table class="scale"> +<caption>Api, New Heb.</caption> + +<tr><td>1.</td><td colspan="2">tai.</td></tr> +<tr><td>2.</td><td colspan="2">lua.</td></tr> +<tr><td>3.</td><td colspan="2">tolu.</td></tr> +<tr><td>4.</td><td colspan="2">vari.</td></tr> +<tr><td>5.</td><td>lima</td><td>= hand.</td></tr> +<tr><td>6.</td><td>o rai</td><td>= other 1.</td></tr> +<tr><td>7.</td><td>o lua</td><td>= other 2.</td></tr> +<tr><td>8.</td><td>o tolo</td><td>= other 3.</td></tr> +<tr><td>9.</td><td>o vari</td><td>= other 4.</td></tr> +<tr><td>10.</td><td>lua lima</td><td>= 2 hands.</td></tr> +</table> + +<table class="scale"> +<caption>Sesake, New Heb.</caption> + +<tr><td>1.</td><td colspan="2">sikai.</td></tr> +<tr><td>2.</td><td colspan="2">dua.</td></tr> +<tr><td>3.</td><td colspan="2">dolu.</td></tr> +<tr><td>4.</td><td colspan="2">pati.</td></tr> +<tr><td>5.</td><td>lima</td><td>= hand.</td></tr> +<tr><td>6.</td><td>la tesa</td><td>= other 1.</td></tr> +<tr><td>7.</td><td>la dua</td><td>= other 2.</td></tr> +<tr><td>8.</td><td>la dolu</td><td>= other 3.</td></tr> +<tr><td>9.</td><td>lo veti</td><td>= other 4.</td></tr> +<tr><td>10.</td><td>dua lima</td><td>= 2 hands.</td></tr> +</table> + +<table class="scale"> +<caption>Pama, New Heb.</caption> + +<tr><td>1.</td><td colspan="2">tai.</td></tr> +<tr><td>2.</td><td colspan="2">e lua.</td></tr> +<tr><td>3.</td><td colspan="2">e tolu.</td></tr> +<tr><td>4.</td><td colspan="2">e hati.</td></tr> +<tr><td>5.</td><td>e lime</td><td>= hand.</td></tr> +<tr><td>6.</td><td>a hitai</td><td>= other 1.</td></tr> +<tr><td>7.</td><td>o lu</td><td>= other 2.</td></tr> +<tr><td>8.</td><td>o tolu</td><td>= other 3.</td></tr> +<tr><td>9.</td><td>o hati</td><td>= other 4.</td></tr> +<tr><td>10.</td><td>ha lua lim</td><td>= 2 hands</td></tr> +</table> + +<table class="scale"> +<caption>Aurora, New Heb.</caption> + +<tr><td>1.</td><td colspan="2">tewa.</td></tr> +<tr><td>2.</td><td colspan="2">i rua.</td></tr> +<tr><td>3.</td><td colspan="2">i tol.</td></tr> +<tr><td>4.</td><td colspan="2">i vat.</td></tr> +<tr><td>5.</td><td>tavalima</td><td>= 1 hand.</td></tr> +<tr><td>6.</td><td>lava tea</td><td>= other 1.</td></tr> +<tr><td>7.</td><td>lava rua</td><td>= other 2.</td></tr> +<tr><td>8.</td><td>lava tol</td><td>= other 3.</td></tr> +<tr><td>9.</td><td>la vat</td><td>= other 4.</td></tr> +<tr><td>10.</td><td>sanwulu</td><td>= two sets.</td></tr> +</table> + + +<table class="scale"> +<caption><a class="pgnum" id="page-156" title="- 156 -"></a>Tobi.<a id="FNA-287" href="#FN-287"><sup>287</sup></a></caption> + +<tr><td>1.</td><td colspan="2">yat.</td></tr> +<tr><td>2.</td><td colspan="2">glu.</td></tr> +<tr><td>3.</td><td colspan="2">ya.</td></tr> +<tr><td>4.</td><td colspan="2">uan.</td></tr> +<tr><td>5.</td><td>yanim</td><td>= 1 hand.</td></tr> +<tr><td>6.</td><td>yawor</td><td>= other 1.</td></tr> +<tr><td>7.</td><td>yavic</td><td>= other 2.</td></tr> +<tr><td>8.</td><td>yawa</td><td>= other 3.</td></tr> +<tr><td>9.</td><td>yatu</td><td>= other 4.</td></tr> +<tr><td>10.</td><td colspan="2">yasec.</td></tr> +</table> + +<table class="scale"> +<caption>Palm Island.<a id="FNA-288" href="#FN-288"><sup>288</sup></a></caption> + +<tr><td>1.</td><td colspan="2">yonkol.</td></tr> +<tr><td>2.</td><td colspan="2">yakka.</td></tr> +<tr><td>3.</td><td colspan="2">tetjora.</td></tr> +<tr><td>4.</td><td colspan="2">tarko.</td></tr> +<tr><td>5.</td><td>yonkol mala</td><td>= 1 hand.</td></tr> +</table> + +<table class="scale"> +<caption>Jajowerong, Victoria.<a href="#FN-288"><sup>288</sup></a></caption> + +<tr><td>1.</td><td colspan="2">kiarp.</td></tr> +<tr><td>2.</td><td colspan="2">bulaits.</td></tr> +<tr><td>3.</td><td>bulaits kiarp</td><td>= 2-1.</td></tr> +<tr><td>4.</td><td>bulaits bulaits</td><td>= 2-2.</td></tr> +<tr><td>5.</td><td>kiarp munnar</td><td>= 1 hand.</td></tr> +<tr><td>6.</td><td>bulaits bulaits bulaits</td><td>= 2-2-2.</td></tr> +<tr><td>10.</td><td>bulaits munnar</td><td>= 2 hands.</td></tr> +</table> + +<p class="noindent">The last two scales deserve special notice. They are +Australian scales, and the former is strongly binary, as +are so many others of that continent. But both show +an incipient quinary tendency in their names for 5 and +10.</p> + +<table class="scale"> +<caption>Cambodia.<a id="FNA-289" href="#FN-289"><sup>289</sup></a></caption> + +<tr><td>1.</td><td colspan="2">muy.</td></tr> +<tr><td>2.</td><td colspan="2">pir.</td></tr> +<tr><td>3.</td><td colspan="2">bey.</td></tr> +<tr><td>4.</td><td colspan="2">buon.</td></tr> +<tr><td>5.</td><td colspan="2">pram.</td></tr> +<tr><td>6.</td><td>pram muy</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>pram pil</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>pram bey</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>pram buon</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">dap.</td></tr> +</table> + +<table class="scale"> +<caption>Tschukschi.<a id="FNA-290" href="#FN-290"><sup>290</sup></a></caption> + +<tr><td>1.</td><td colspan="2">inen.</td></tr> +<tr><td>2.</td><td colspan="2">nirach.</td></tr> +<tr><td>3.</td><td colspan="2">n'roch.</td></tr> +<tr><td>4.</td><td colspan="2">n'rach.</td></tr> +<tr><td>5.</td><td>miligen</td><td>= hand.</td></tr> +<tr><td>6.</td><td>inen miligen</td><td>= 1-5.</td></tr> +<tr><td>7.</td><td>nirach miligen</td><td>= 2-5.</td></tr> +<tr><td>8.</td><td colspan="2">anwrotkin.</td></tr> +<tr><td>9.</td><td colspan="2">chona tsinki.</td></tr> +<tr><td>10.</td><td>migitken</td><td>= both hands.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-157" title="- 157 -"></a>Kottisch<a id="FNA-291" href="#FN-291"><sup>291</sup></a></caption> + +<tr><td>1.</td><td colspan="2">hutsa.</td></tr> +<tr><td>2.</td><td colspan="2">ina.</td></tr> +<tr><td>3.</td><td colspan="2">tona.</td></tr> +<tr><td>4.</td><td colspan="2">sega.</td></tr> +<tr><td>5.</td><td colspan="2">chega.</td></tr> +<tr><td>6.</td><td>chelutsa</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>chelina</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>chaltona</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>tsumnaga</td><td>= 10 − 1.</td></tr> +<tr><td>10.</td><td colspan="2">haga.</td></tr> +</table> + +<table class="scale"> +<caption>Eskimo of N.-W. Alaska.<a id="FNA-292" href="#FN-292"><sup>292</sup></a></caption> + +<tr><td>1.</td><td colspan="2">a towshek.</td></tr> +<tr><td>2.</td><td colspan="2">hipah, or malho.</td></tr> +<tr><td>3.</td><td colspan="2">pingishute.</td></tr> +<tr><td>4.</td><td colspan="2">sesaimat.</td></tr> +<tr><td>5.</td><td colspan="2">talema.</td></tr> +<tr><td>6.</td><td>okvinile, or ahchegaret</td><td>= another 1?</td></tr> +<tr><td>7.</td><td>talema-malronik</td><td>= 5-two of them.</td></tr> +<tr><td>8.</td><td>pingishu-okvingile</td><td>= 2d 3?</td></tr> +<tr><td>9.</td><td>kolingotalia</td><td>= 10 − 1?</td></tr> +<tr><td>10.</td><td colspan="2">koleet.</td></tr> +</table> + +<table class="scale"> +<caption>Kamtschatka, South.<a id="FNA-293" href="#FN-293"><sup>293</sup></a></caption> + +<tr><td>1.</td><td colspan="2">dischak.</td></tr> +<tr><td>2.</td><td colspan="2">kascha.</td></tr> +<tr><td>3.</td><td colspan="2">tschook.</td></tr> +<tr><td>4.</td><td colspan="2">tschaaka.</td></tr> +<tr><td>5.</td><td colspan="2">kumnaka.</td></tr> +<tr><td>6.</td><td colspan="2">ky'lkoka.</td></tr> +<tr><td>7.</td><td>itatyk</td><td>= 2 + 5.</td></tr> +<tr><td>8.</td><td>tschookotuk</td><td>= 3 + 5.</td></tr> +<tr><td>9.</td><td>tschuaktuk</td><td>= 4 + 5.</td></tr> +<tr><td>10.</td><td>kumechtuk</td><td>= 5 + 5.</td></tr> +</table> + +<table class="scale"> +<caption>Aleuts<a id="FNA-294" href="#FN-294"><sup>294</sup></a></caption> + +<tr><td>1.</td><td colspan="2">ataqan.</td></tr> +<tr><td>2.</td><td colspan="2">aljak.</td></tr> +<tr><td>3.</td><td colspan="2">qankun.</td></tr> +<tr><td>4.</td><td colspan="2">sitsin.</td></tr> +<tr><td>5.</td><td>tsan</td><td>= my hand.</td></tr> +<tr><td>6.</td><td>atun</td><td>= 1 + 5.</td></tr> +<tr><td>7.</td><td>ulun</td><td>= 2 + 5.</td></tr> +<tr><td>8.</td><td>qamtsin</td><td>= 3 + 5.</td></tr> +<tr><td>9.</td><td>sitsin</td><td>= 4 + 5.</td></tr> +<tr><td>10.</td><td colspan="2">hatsiq.</td></tr> +</table> + +<table class="scale"> +<caption>Tchiglit, Mackenzie R.<a id="FNA-295" href="#FN-295"><sup>295</sup></a></caption> + +<tr><td>1.</td><td colspan="2">ataotçirkr.</td></tr> +<tr><td>2.</td><td colspan="2">aypak, or malloerok.</td></tr> +<tr><td>3.</td><td colspan="2">illaak, or piñatcut.</td></tr> +<tr><td>4.</td><td colspan="2">tçitamat.</td></tr> +<tr><td>5.</td><td colspan="2">tallemat.</td></tr> +<tr><td>6.</td><td colspan="2">arveneloerit.</td></tr> +<tr><td>7.</td><td>arveneloerit-aypak</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>arveneloerit-illaak</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>arveneloerit-tçitamat</td><td>= 5 + 4.</td></tr> +<tr><td>10.</td><td colspan="2">krolit.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-158" title="- 158 -"></a>Sahaptin (Nez Perces).<a id="FNA-296" href="#FN-296"><sup>296</sup></a></caption> + +<tr><td>1.</td><td colspan="2">naks.</td></tr> +<tr><td>2.</td><td colspan="2">lapit.</td></tr> +<tr><td>3.</td><td colspan="2">mitat.</td></tr> +<tr><td>4.</td><td>pi-lapt</td><td>= 2 × 2.</td></tr> +<tr><td>5.</td><td colspan="2">pachat.</td></tr> +<tr><td>6.</td><td>oi-laks</td><td>= [5] + 1.</td></tr> +<tr><td>7.</td><td>oi-napt</td><td>= [5] + 2.</td></tr> +<tr><td>8.</td><td>oi-matat</td><td>= [5] + 3.</td></tr> +<tr><td>9.</td><td colspan="2">koits.</td></tr> +<tr><td>10.</td><td colspan="2">putimpt.</td></tr> +</table> + +<table class="scale"> +<caption>Greenland.<a id="FNA-297" href="#FN-297"><sup>297</sup></a></caption> + +<tr><td>1.</td><td colspan="2">atauseq.</td></tr> +<tr><td>2.</td><td colspan="2">machdluq.</td></tr> +<tr><td>3.</td><td colspan="2">pinasut.</td></tr> +<tr><td>4.</td><td colspan="2">sisamat</td></tr> +<tr><td>5.</td><td colspan="2">tadlimat.</td></tr> +<tr><td>6.</td><td>achfineq-atauseq</td><td>= other hand 1.</td></tr> +<tr><td>7.</td><td>achfineq-machdluq</td><td>= other hand 2.</td></tr> +<tr><td>8.</td><td>achfineq-pinasut</td><td>= other hand 3.</td></tr> +<tr><td>9.</td><td>achfineq-sisamat</td><td>= other hand 4.</td></tr> +<tr><td>10.</td><td colspan="2">qulit.</td></tr> +<tr><td>11.</td><td>achqaneq-atauseq</td><td>= first foot 1.</td></tr> +<tr><td>12.</td><td>achqaneq-machdluq</td><td>= first foot 2.</td></tr> +<tr><td>13.</td><td>achqaneq-pinasut</td><td>= first foot 3.</td></tr> +<tr><td>14.</td><td>achqaneq-sisamat</td><td>= first foot 4.</td></tr> +<tr><td>15.</td><td colspan="2">achfechsaneq?</td></tr> +<tr><td>16.</td><td>achfechsaneq-atauseq</td><td>= other foot 1.</td></tr> +<tr><td>17.</td><td>achfechsaneq-machdlup</td><td>= other foot 2.</td></tr> +<tr><td>18.</td><td>achfechsaneq-pinasut</td><td>= other foot 3.</td></tr> +<tr><td>19.</td><td>achfechsaneq-sisamat</td><td>= other foot 4.</td></tr> +<tr><td>20.</td><td>inuk navdlucho</td><td>= a man ended.</td></tr> +</table> + +<p>Up to this point the Greenlander's scale is almost +purely quinary. Like those of which mention was made +at the beginning of this chapter, it persists in progressing +by fives until it reaches 20, when it announces a +new base, which shows that the system will from now +on be vigesimal. This scale is one of the most interesting +of which we have any record, and will be +<a class="pgnum" id="page-159" title="- 159 -"></a>noticed again in the next chapter. In many respects +it is like the scale of the Point Barrow Eskimo, which +was given early in Chapter III. The Eskimo languages +are characteristically quinary-vigesimal in their number +systems, but few of them present such perfect +examples of that method of counting as do the two +just mentioned.</p> + +<table class="scale"> +<caption>Chippeway.<a id="FNA-298" href="#FN-298"><sup>298</sup></a></caption> + +<tr><td>1.</td><td colspan="2">bejig.</td></tr> +<tr><td>2.</td><td colspan="2">nij.</td></tr> +<tr><td>3.</td><td colspan="2">nisswi.</td></tr> +<tr><td>4.</td><td colspan="2">niwin.</td></tr> +<tr><td>5.</td><td colspan="2">nanun.</td></tr> +<tr><td>6.</td><td>ningotwasswi</td><td>= 1 again?</td></tr> +<tr><td>7.</td><td>nijwasswi</td><td>= 2 again?</td></tr> +<tr><td>8.</td><td>nishwasswi</td><td>= 3 again?</td></tr> +<tr><td>9.</td><td>jangasswi</td><td>= 4 again?</td></tr> +<tr><td>10.</td><td>midasswi</td><td>= 5 again.</td></tr> +</table> + +<table class="scale"> +<caption>Massachusetts.<a id="FNA-299" href="#FN-299"><sup>299</sup></a></caption> + +<tr><td>1.</td><td colspan="2">nequt.</td></tr> +<tr><td>2.</td><td colspan="2">neese.</td></tr> +<tr><td>3.</td><td colspan="2">nish.</td></tr> +<tr><td>4.</td><td colspan="2">yaw.</td></tr> +<tr><td>5.</td><td>napanna</td><td>= on one side, <i>i.e.</i> 1 hand.</td></tr> +<tr><td>6.</td><td>nequttatash</td><td>= 1 added.</td></tr> +<tr><td>7.</td><td>nesausuk</td><td>= 2 again?</td></tr> +<tr><td>8.</td><td>shawosuk</td><td>= 3 again?</td></tr> +<tr><td>9.</td><td>pashoogun</td><td>= it comes near, <i>i.e.</i> to 10.</td></tr> +<tr><td>10.</td><td colspan="2">puik.</td></tr> +</table> + +<table class="scale"> +<caption>Ojibwa of Chegoimegon.<a id="FNA-300" href="#FN-300"><sup>300</sup></a></caption> + +<tr><td>1.</td><td colspan="2">bashik.</td></tr> +<tr><td>2.</td><td colspan="2">neensh.</td></tr> +<tr><td>3.</td><td colspan="2">niswe.</td></tr> +<tr><td>4.</td><td colspan="2">newin.</td></tr> +<tr><td>5.</td><td colspan="2">nanun.</td></tr> +<tr><td>6.</td><td>ningodwaswe</td><td>= 1 again?</td></tr> +<tr><td>7.</td><td>nishwaswe</td><td>= 2 again?</td></tr> +<tr><td>8.</td><td>shouswe</td><td>= 3 again?</td></tr> +<tr><td>9.</td><td>shangaswe</td><td>= 4 again?</td></tr> +<tr><td>10.</td><td>medaswe</td><td>= 5 again?</td></tr> +</table> + +<table class="scale"> +<caption>Ottawa.</caption> + +<tr><td>1.</td><td colspan="2">ningotchau.</td></tr> +<tr><td>2.</td><td colspan="2">ninjwa.</td></tr> +<tr><td>3.</td><td colspan="2">niswa.</td></tr> +<tr><td>4.</td><td colspan="2">niwin.</td></tr> +<tr><td>5.</td><td colspan="2">nanau.</td></tr> +<tr><td>6.</td><td>ningotwaswi</td><td>= 1 again?</td></tr> +<tr><td>7.</td><td>ninjwaswi</td><td>= 2 again?</td></tr> +<tr><td>8.</td><td>nichwaswi</td><td>= 3 again?</td></tr> +<tr><td>9.</td><td colspan="2">shang.</td></tr> +<tr><td>10.</td><td colspan="2">kwetch.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-160" title="- 160 -"></a>Delaware.</caption> + +<tr><td>1.</td><td colspan="2">n'gutti.</td></tr> +<tr><td>2.</td><td colspan="2">niskha.</td></tr> +<tr><td>3.</td><td colspan="2">nakha.</td></tr> +<tr><td>4.</td><td colspan="2">newa.</td></tr> +<tr><td>5.</td><td colspan="2">nalan [akin to palenach, hand].</td></tr> +<tr><td>6.</td><td>guttash</td><td>= 1 on the other side.</td></tr> +<tr><td>7.</td><td>nishash</td><td>= 2 on the other side.</td></tr> +<tr><td>8.</td><td>khaash</td><td>= 3 on the other side.</td></tr> +<tr><td>9.</td><td>peshgonk</td><td>= coming near.</td></tr> +<tr><td>10.</td><td>tellen</td><td>= no more.</td></tr> +</table> + +<table class="scale"> +<caption>Shawnoe.</caption> + +<tr><td>1.</td><td colspan="2">negote.</td></tr> +<tr><td>2.</td><td colspan="2">neshwa.</td></tr> +<tr><td>3.</td><td colspan="2">nithuie.</td></tr> +<tr><td>4.</td><td colspan="2">newe.</td></tr> +<tr><td>5.</td><td>nialinwe</td><td>= gone.</td></tr> +<tr><td>6.</td><td>negotewathwe</td><td>= 1 further.</td></tr> +<tr><td>7.</td><td>neshwathwe</td><td>= 2 further.</td></tr> +<tr><td>8.</td><td>sashekswa</td><td>= 3 further?</td></tr> +<tr><td>9.</td><td colspan="2">chakatswe [akin to chagisse, “used up”].</td></tr> +<tr><td>10.</td><td>metathwe</td><td>= no further.</td></tr> +</table> + +<table class="scale"> +<caption>Micmac.<a id="FNA-301" href="#FN-301"><sup>301</sup></a></caption> + +<tr><td>1.</td><td>naiookt.</td></tr> +<tr><td>2.</td><td>tahboo.</td></tr> +<tr><td>3.</td><td>seest.</td></tr> +<tr><td>4.</td><td>naioo.</td></tr> +<tr><td>5.</td><td>nahn.</td></tr> +<tr><td>6.</td><td>usoo-cum.</td></tr> +<tr><td>7.</td><td>eloo-igunuk.</td></tr> +<tr><td>8.</td><td>oo-gumoolchin.</td></tr> +<tr><td>9.</td><td>pescoonaduk.</td></tr> +<tr><td>10.</td><td>mtlin.</td></tr> +</table> + +<p>One peculiarity of the Micmac numerals is most noteworthy. +The numerals are real verbs, instead of adjectives, +or, as is sometimes the case, nouns. They are +conjugated through all the variations of mood, tense, person, +and number. The forms given above are not those +that would be used in counting, but are for specific use, +being varied according to the thought it was intended +to express. For example, <i>naiooktaich</i> = there is 1, is +present tense; <i>naiooktaichcus</i>, there was 1, is imperfect; +and <i>encoodaichdedou</i>, there will be 1, is future. +The variation in person is shown by the following +inflection:</p> + +<table class="grammar"> +<caption><a class="pgnum" id="page-161" title="- 161 -"></a>Present Tense.</caption> + +<tr><td>1st pers.</td><td>tahboosee-ek</td><td>= there are 2 of us.</td></tr> +<tr><td>2d pers.</td><td>tahboosee-yok</td><td>= there are 2 of you.</td></tr> +<tr><td>3d pers.</td><td>tahboo-sijik</td><td>= there are 2 of them.</td></tr> +</table> + +<table class="grammar"> +<caption>Imperfect Tense.</caption> + +<tr><td>1st pers.</td><td>tahboosee-egup</td><td>= there were 2 of us.</td></tr> +<tr><td>2d pers.</td><td>tahboosee-yogup</td><td>= there were 2 of you.</td></tr> +<tr><td>3d pers.</td><td>tahboosee-sibunik</td><td>= there were 2 of them.</td></tr> +</table> + +<table class="grammar"> +<caption>Future Tense.</caption> + +<tr><td>3d pers.</td><td>tahboosee-dak</td><td>= there will be 2 of them, etc.</td></tr> +</table> + +<p>The negative form is also comprehended in the list +of possible variations. Thus, <i>tahboo-seekw</i>, there are not +2 of them; <i>mah tahboo-seekw</i>, there will not be 2 of +them; and so on, through all the changes which the conjugation +of the verb permits.</p> + +<table class="scale"> +<caption>Old Algonquin.</caption> + +<tr><td>1.</td><td colspan="2">peygik.</td></tr> +<tr><td>2.</td><td colspan="2">ninsh.</td></tr> +<tr><td>3.</td><td colspan="2">nisswey.</td></tr> +<tr><td>4.</td><td colspan="2">neyoo.</td></tr> +<tr><td>5.</td><td>nahran</td><td>= gone.</td></tr> +<tr><td>6.</td><td>ningootwassoo</td><td>= 1 on the other side.</td></tr> +<tr><td>7.</td><td>ninshwassoo</td><td>= 2 on the other side.</td></tr> +<tr><td>8.</td><td>nisswasso</td><td>= 3 on the other side.</td></tr> +<tr><td>9.</td><td colspan="2">shangassoo [akin to chagisse, “used up”].</td></tr> +<tr><td>10.</td><td>mitassoo</td><td>= no further.</td></tr> +</table> + +<table class="scale"> +<caption>Omaha.</caption> + +<tr><td>1.</td><td colspan="2">meeachchee.</td></tr> +<tr><td>2.</td><td colspan="2">nomba.</td></tr> +<tr><td>3.</td><td colspan="2">rabeenee.</td></tr> +<tr><td>4.</td><td colspan="2">tooba.</td></tr> +<tr><td>5.</td><td>satta</td><td>= hand, <i>i.e.</i> all the fingers turned down.</td></tr> +<tr><td>6.</td><td>shappai</td><td>= 1 more.</td></tr> +<tr><td>7.</td><td>painumba</td><td>= fingers 2.</td></tr> +<tr><td>8.</td><td>pairabeenee</td><td>= fingers 3.</td></tr> +<tr><td>9.</td><td>shonka</td><td>= only 1 finger (remains).</td></tr> +<tr><td>10.</td><td>kraibaira</td><td>= unbent.<a id="FNA-302" href="#FN-302"><sup>302</sup></a></td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-162" title="- 162 -"></a>Choctaw.</caption> + +<tr><td>1.</td><td colspan="2">achofee.</td></tr> +<tr><td>2.</td><td colspan="2">tuklo.</td></tr> +<tr><td>3.</td><td colspan="2">tuchina.</td></tr> +<tr><td>4.</td><td colspan="2">ushta.</td></tr> +<tr><td>5.</td><td>tahlape</td><td>= the first hand ends.</td></tr> +<tr><td>6.</td><td colspan="2">hanali.</td></tr> +<tr><td>7.</td><td>untuklo</td><td>= again 2.</td></tr> +<tr><td>8.</td><td>untuchina</td><td>= again 3.</td></tr> +<tr><td>9.</td><td>chokali</td><td>= soon the end; <i>i.e.</i> next the last.</td></tr> +<tr><td>10.</td><td colspan="2">pokoli.</td></tr> +</table> + +<table class="scale"> +<caption>Caddoe.</caption> + +<tr><td>1.</td><td colspan="2">kouanigh.</td></tr> +<tr><td>2.</td><td colspan="2">behit.</td></tr> +<tr><td>3.</td><td colspan="2">daho.</td></tr> +<tr><td>4.</td><td colspan="2">hehweh.</td></tr> +<tr><td>5.</td><td colspan="2">dihsehkon.</td></tr> +<tr><td>6.</td><td colspan="2">dunkeh.</td></tr> +<tr><td>7.</td><td>bisekah</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>dousehka</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>hehwehsehka</td><td>= 4 + hand.</td></tr> +<tr><td>10.</td><td colspan="2">behnehaugh.</td></tr> +</table> + +<table class="scale"> +<caption>Chippeway.</caption> + +<tr><td>1.</td><td colspan="2">payshik.</td></tr> +<tr><td>2.</td><td colspan="2">neesh.</td></tr> +<tr><td>3.</td><td colspan="2">neeswoy.</td></tr> +<tr><td>4.</td><td colspan="2">neon.</td></tr> +<tr><td>5.</td><td>naman</td><td>= gone.</td></tr> +<tr><td>6.</td><td>nequtwosswoy</td><td>= 1 on the other side.</td></tr> +<tr><td>7.</td><td>neeshswosswoy</td><td>= 2 on the other side.</td></tr> +<tr><td>8.</td><td>swoswoy</td><td>= 3 on the other side?</td></tr> +<tr><td>9.</td><td colspan="2">shangosswoy [akin to chagissi, “used up”].</td></tr> +<tr><td>10.</td><td>metosswoy</td><td>= no further.</td></tr> +</table> + +<table class="scale"> +<caption>Adaize.</caption> + +<tr><td>1.</td><td colspan="2">nancas.</td></tr> +<tr><td>2.</td><td colspan="2">nass.</td></tr> +<tr><td>3.</td><td colspan="2">colle.</td></tr> +<tr><td>4.</td><td colspan="2">tacache.</td></tr> +<tr><td>5.</td><td colspan="2">seppacan.</td></tr> +<tr><td>6.</td><td>pacanancus</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>pacaness</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>pacalcon</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>sickinish</td><td>= hands minus?</td></tr> +<tr><td>10.</td><td colspan="2">neusne.</td></tr> +</table> + +<table class="scale"> +<caption>Pawnee.</caption> + +<tr><td>1.</td><td colspan="2">askoo.</td></tr> +<tr><td>2.</td><td colspan="2">peetkoo.</td></tr> +<tr><td>3.</td><td colspan="2">touweet.</td></tr> +<tr><td>4.</td><td colspan="2">shkeetiksh.</td></tr> +<tr><td>5.</td><td>sheeooksh</td><td>= hands half.</td></tr> +<tr><td>6.</td><td>sheekshabish</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>peetkoosheeshabish</td><td>= 2 + 5.</td></tr> +<tr><td>8.</td><td>touweetshabish</td><td>= 3 + 5.</td></tr> +<tr><td>9.</td><td>looksheereewa</td><td>= 10 − 1.</td></tr> +<tr><td>10.</td><td>looksheeree</td><td>= 2d 5?</td></tr> +</table> + +<table class="scale"> +<caption>Minsi.</caption> + +<tr><td>1.</td><td colspan="2">gutti.</td></tr> +<tr><td>2.</td><td colspan="2">niskha.</td></tr> +<tr><td>3.</td><td colspan="2">nakba.</td></tr> +<tr><td>4.</td><td colspan="2">newa.</td></tr> +<tr><td>5.</td><td>nulan</td><td>= gone?</td></tr> +<tr><td>6.</td><td>guttash</td><td>= 1 added.</td></tr> +<tr><td>7.</td><td>nishoash</td><td>= 2 added.</td></tr> +<tr><td>8.</td><td>khaash</td><td>= 3 added.</td></tr> +<tr><td>9.</td><td colspan="2">noweli.</td></tr> +<tr><td>10.</td><td colspan="2">wimbat.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-163" title="- 163 -"></a>Konlischen.</caption> + +<tr><td>1.</td><td colspan="2">tlek.</td></tr> +<tr><td>2.</td><td colspan="2">tech.</td></tr> +<tr><td>3.</td><td colspan="2">nezk.</td></tr> +<tr><td>4.</td><td colspan="2">taakun.</td></tr> +<tr><td>5.</td><td colspan="2">kejetschin.</td></tr> +<tr><td>6.</td><td>klet uschu</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>tachate uschu</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>nesket uschu</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>kuschok</td><td>= 10 − 1?</td></tr> +<tr><td>10.</td><td colspan="2">tschinkat.</td></tr> +</table> + +<table class="scale"> +<caption>Tlingit.<a id="FNA-303" href="#FN-303"><sup>303</sup></a></caption> + +<tr><td>1.</td><td colspan="2">tlek.</td></tr> +<tr><td>2.</td><td colspan="2">deq.</td></tr> +<tr><td>3.</td><td colspan="2">natsk.</td></tr> +<tr><td>4.</td><td>dak'on</td><td>= 2d 2.</td></tr> +<tr><td>5.</td><td>kedjin</td><td>= hand.</td></tr> +<tr><td>6.</td><td>tle durcu</td><td>= other 1.</td></tr> +<tr><td>7.</td><td>daqa durcu</td><td>= other 2.</td></tr> +<tr><td>8.</td><td>natska durcu</td><td>= other 3.</td></tr> +<tr><td>9.</td><td colspan="2">gocuk.</td></tr> +<tr><td>10.</td><td>djinkat</td><td>= both hands.</td></tr> +</table> + +<table class="scale"> +<caption>Rapid, or Fall, Indians.</caption> + +<tr><td>1.</td><td colspan="2">karci.</td></tr> +<tr><td>2.</td><td colspan="2">neece.</td></tr> +<tr><td>3.</td><td colspan="2">narce.</td></tr> +<tr><td>4.</td><td colspan="2">nean.</td></tr> +<tr><td>5.</td><td colspan="2">yautune.</td></tr> +<tr><td>6.</td><td>neteartuce</td><td>= 1 over?</td></tr> +<tr><td>7.</td><td>nesartuce</td><td>= 2 over?</td></tr> +<tr><td>8.</td><td>narswartuce</td><td>= 3 over?</td></tr> +<tr><td>9.</td><td>anharbetwartuce</td><td>= 4 over?</td></tr> +<tr><td>10.</td><td>mettartuce</td><td>= no further?</td></tr> +</table> + +<table class="scale"> +<caption>Heiltsuk.<a id="FNA-304" href="#FN-304"><sup>304</sup></a></caption> + +<tr><td>1.</td><td colspan="2">men.</td></tr> +<tr><td>2.</td><td colspan="2">matl.</td></tr> +<tr><td>3.</td><td colspan="2">yutq.</td></tr> +<tr><td>4.</td><td colspan="2">mu.</td></tr> +<tr><td>5.</td><td colspan="2">sky'a.</td></tr> +<tr><td>6.</td><td colspan="2">katla.</td></tr> +<tr><td>7.</td><td>matlaaus</td><td>= other 2?</td></tr> +<tr><td>8.</td><td>yutquaus</td><td>= other 3?</td></tr> +<tr><td>9.</td><td>mamene</td><td>= 10 − 1.</td></tr> +<tr><td>10.</td><td colspan="2">aiky'as.</td></tr> +</table> + +<table class="scale"> +<caption>Nootka.<a id="FNA-305" href="#FN-305"><sup>305</sup></a></caption> + +<tr><td>1.</td><td colspan="2">nup.</td></tr> +<tr><td>2.</td><td colspan="2">atla.</td></tr> +<tr><td>3.</td><td colspan="2">katstsa.</td></tr> +<tr><td>4.</td><td colspan="2">mo.</td></tr> +<tr><td>5.</td><td colspan="2">sutca.</td></tr> +<tr><td>6.</td><td>nopo</td><td>= other 1?</td></tr> +<tr><td>7.</td><td>atlpo</td><td>= other 2?</td></tr> +<tr><td>8.</td><td>atlakutl</td><td>= 10 − 2.</td></tr> +<tr><td>9.</td><td>ts'owakutl</td><td>= 10 − 1.</td></tr> +<tr><td>10.</td><td colspan="2">haiu.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-164" title="- 164 -"></a>Tsimshian.<a id="FNA-306" href="#FN-306"><sup>306</sup></a></caption> + +<tr><td>1.</td><td colspan="2">gyak.</td></tr> +<tr><td>2.</td><td colspan="2">tepqat.</td></tr> +<tr><td>3.</td><td colspan="2">guant.</td></tr> +<tr><td>4.</td><td colspan="2">tqalpq.</td></tr> +<tr><td>5.</td><td colspan="2">kctonc (from <i>anon</i>, hand).</td></tr> +<tr><td>6.</td><td>kalt</td><td>= 2d 1.</td></tr> +<tr><td>7.</td><td>t'epqalt</td><td>= 2d 2.</td></tr> +<tr><td>8.</td><td>guandalt</td><td>= 2d 3?</td></tr> +<tr><td>9.</td><td colspan="2">kctemac.</td></tr> +<tr><td>10.</td><td colspan="2">gy'ap.</td></tr> +</table> + +<table class="scale"> +<caption>Bilqula.<a href="#FN-306"><sup>306</sup></a></caption> + +<tr><td>1.</td><td colspan="2">(s)maotl.</td></tr> +<tr><td>2.</td><td colspan="2">tlnos.</td></tr> +<tr><td>3.</td><td colspan="2">asmost.</td></tr> +<tr><td>4.</td><td colspan="2">mos.</td></tr> +<tr><td>5.</td><td colspan="2">tsech.</td></tr> +<tr><td>6.</td><td>tqotl</td><td>= 2d 1?</td></tr> +<tr><td>7.</td><td>nustlnos</td><td>= 2d 2?</td></tr> +<tr><td>8.</td><td>k'etlnos</td><td>= 2 × 4.</td></tr> +<tr><td>9.</td><td colspan="2">k'esman.</td></tr> +<tr><td>10.</td><td colspan="2">tskchlakcht.</td></tr> +</table> + +<table class="scale"> +<caption>Molele.<a id="FNA-307" href="#FN-307"><sup>307</sup></a></caption> + +<tr><td>1.</td><td colspan="2">mangu.</td></tr> +<tr><td>2.</td><td colspan="2">lapku.</td></tr> +<tr><td>3.</td><td colspan="2">mutka.</td></tr> +<tr><td>4.</td><td colspan="2">pipa.</td></tr> +<tr><td>5.</td><td colspan="2">pika.</td></tr> +<tr><td>6.</td><td>napitka</td><td>= 1 + 5.</td></tr> +<tr><td>7.</td><td>lapitka</td><td>= 2 + 5.</td></tr> +<tr><td>8.</td><td>mutpitka</td><td>= 3 + 5.</td></tr> +<tr><td>9.</td><td colspan="2">laginstshiatkus.</td></tr> +<tr><td>10.</td><td colspan="2">nawitspu.</td></tr> +</table> + +<table class="scale"> +<caption>Waiilatpu.<a id="FNA-308" href="#FN-308"><sup>308</sup></a></caption> + +<tr><td>1.</td><td colspan="2">na.</td></tr> +<tr><td>2.</td><td colspan="2">leplin.</td></tr> +<tr><td>3.</td><td colspan="2">matnin.</td></tr> +<tr><td>4.</td><td colspan="2">piping.</td></tr> +<tr><td>5.</td><td colspan="2">tawit.</td></tr> +<tr><td>6.</td><td>noina</td><td>= [5] + 1.</td></tr> +<tr><td>7.</td><td>noilip</td><td>= [5] + 2.</td></tr> +<tr><td>8.</td><td>noimat</td><td>= [5] + 3.</td></tr> +<tr><td>9.</td><td colspan="2">tanauiaishimshim.</td></tr> +<tr><td>10.</td><td colspan="2">ningitelp.</td></tr> +</table> + +<table class="scale"> +<caption>Lutuami.<a href="#FN-307"><sup>307</sup></a></caption> + +<tr><td>1.</td><td colspan="2">natshik.</td></tr> +<tr><td>2.</td><td colspan="2">lapit.</td></tr> +<tr><td>3.</td><td colspan="2">ntani.</td></tr> +<tr><td>4.</td><td colspan="2">wonip.</td></tr> +<tr><td>5.</td><td colspan="2">tonapni.</td></tr> +<tr><td>6.</td><td>nakskishuptane</td><td>= 1 + 5.</td></tr> +<tr><td>7.</td><td>tapkishuptane</td><td>= 2 + 5.</td></tr> +<tr><td>8.</td><td>ndanekishuptane</td><td>= 3 + 5.</td></tr> +<tr><td>9.</td><td>natskaiakish</td><td>= 10 − 1.</td></tr> +<tr><td>10.</td><td colspan="2">taunip.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-165" title="- 165 -"></a>Saste (Shasta).<a id="FNA-309" href="#FN-309"><sup>309</sup></a></caption> + +<tr><td>1.</td><td colspan="2">tshiamu.</td></tr> +<tr><td>2.</td><td colspan="2">hoka.</td></tr> +<tr><td>3.</td><td colspan="2">hatski.</td></tr> +<tr><td>4.</td><td colspan="2">irahaia.</td></tr> +<tr><td>5.</td><td colspan="2">etsha.</td></tr> +<tr><td>6.</td><td colspan="2">tahaia.</td></tr> +<tr><td>7.</td><td>hokaikinis</td><td>= 2 + 5.</td></tr> +<tr><td>8.</td><td>hatsikikiri</td><td>= 3 + 5.</td></tr> +<tr><td>9.</td><td colspan="2">kirihariki-ikiriu.</td></tr> +<tr><td>10.</td><td colspan="2">etsehewi.</td></tr> +</table> + +<table class="scale"> +<caption>Cahuillo.<a id="FNA-310" href="#FN-310"><sup>310</sup></a></caption> + +<tr><td>1.</td><td colspan="2">supli.</td></tr> +<tr><td>2.</td><td colspan="2">mewi.</td></tr> +<tr><td>3.</td><td colspan="2">mepai.</td></tr> +<tr><td>4.</td><td colspan="2">mewittsu.</td></tr> +<tr><td>5.</td><td colspan="2">nomekadnun.</td></tr> +<tr><td>6.</td><td>kadnun-supli</td><td>= 5-1.</td></tr> +<tr><td>7.</td><td>kan-munwi</td><td>= 5-2.</td></tr> +<tr><td>8.</td><td>kan-munpa</td><td>= 5-3.</td></tr> +<tr><td>9.</td><td>kan-munwitsu</td><td>= 5-4.</td></tr> +<tr><td>10.</td><td colspan="2">nomatsumi.</td></tr> +</table> + +<table class="scale"> +<caption>Timukua.<a id="FNA-311" href="#FN-311"><sup>311</sup></a></caption> + +<tr><td>1.</td><td colspan="2">yaha.</td></tr> +<tr><td>2.</td><td colspan="2">yutsa.</td></tr> +<tr><td>3.</td><td colspan="2">hapu.</td></tr> +<tr><td>4.</td><td colspan="2">tseketa.</td></tr> +<tr><td>5.</td><td colspan="2">marua.</td></tr> +<tr><td>6.</td><td>mareka</td><td>= 5 + 1</td></tr> +<tr><td>7.</td><td>pikitsa</td><td>= 5 + 2</td></tr> +<tr><td>8.</td><td>pikinahu</td><td>= 5 + 3</td></tr> +<tr><td>9.</td><td>peke-tsaketa</td><td>= 5 + 4</td></tr> +<tr><td>10.</td><td colspan="2">tuma.</td></tr> +</table> + +<table class="scale"> +<caption>Otomi<a id="FNA-312" href="#FN-312"><sup>312</sup></a></caption> + +<tr><td>1.</td><td colspan="2">nara.</td></tr> +<tr><td>2.</td><td colspan="2">yocho.</td></tr> +<tr><td>3.</td><td colspan="2">chiu.</td></tr> +<tr><td>4.</td><td colspan="2">gocho.</td></tr> +<tr><td>5.</td><td colspan="2">kuto.</td></tr> +<tr><td>6.</td><td>rato</td><td>= 1 + 5.</td></tr> +<tr><td>7.</td><td>yoto</td><td>= 2 + 5.</td></tr> +<tr><td>8.</td><td>chiato</td><td>= 3 + 5.</td></tr> +<tr><td>9.</td><td>guto</td><td>= 4 + 5.</td></tr> +<tr><td>10.</td><td colspan="2">reta.</td></tr> +</table> + +<table class="scale"> +<caption>Tarasco.<a id="FNA-313" href="#FN-313"><sup>313</sup></a></caption> + +<tr><td>1.</td><td colspan="2">ma.</td></tr> +<tr><td>2.</td><td colspan="2">dziman.</td></tr> +<tr><td>3.</td><td colspan="2">tanimo.</td></tr> +<tr><td>4.</td><td colspan="2">tamu.</td></tr> +<tr><td>5.</td><td colspan="2">yumu.</td></tr> +<tr><td>6.</td><td colspan="2">kuimu.</td></tr> +<tr><td>7.</td><td>yun-dziman</td><td>= [5] + 2.</td></tr> +<tr><td>8.</td><td>yun-tanimo</td><td>= [5] + 3.</td></tr> +<tr><td>9.</td><td>yun-tamu</td><td>= [5] + 4.</td></tr> +<tr><td>10.</td><td colspan="2">temben.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-166" title="- 166 -"></a>Matlaltzincan.<a id="FNA-314" href="#FN-314"><sup>314</sup></a></caption> + +<tr><td>1.</td><td colspan="2">indawi.</td></tr> +<tr><td>2.</td><td colspan="2">inawi.</td></tr> +<tr><td>3.</td><td colspan="2">inyuhu.</td></tr> +<tr><td>4.</td><td colspan="2">inkunowi.</td></tr> +<tr><td>5.</td><td colspan="2">inkutaa.</td></tr> +<tr><td>6.</td><td>inda-towi</td><td>= 1 + 5.</td></tr> +<tr><td>7.</td><td>ine-towi</td><td>= 2 + 5.</td></tr> +<tr><td>8.</td><td>ine-ukunowi</td><td>= 2-4.</td></tr> +<tr><td>9.</td><td>imuratadahata</td><td>= 10 − 1?</td></tr> +<tr><td>10.</td><td colspan="2">inda-hata.</td></tr> +</table> + +<table class="scale"> +<caption>Cora.<a id="FNA-315" href="#FN-315"><sup>315</sup></a></caption> + +<tr><td>1.</td><td colspan="2">ceaut.</td></tr> +<tr><td>2.</td><td colspan="2">huapoa.</td></tr> +<tr><td>3.</td><td colspan="2">huaeica.</td></tr> +<tr><td>4.</td><td colspan="2">moacua.</td></tr> +<tr><td>5.</td><td colspan="2">anxuvi.</td></tr> +<tr><td>6.</td><td>a-cevi</td><td>= [5] + 1.</td></tr> +<tr><td>7.</td><td>a-huapoa</td><td>= [5] + 2.</td></tr> +<tr><td>8.</td><td>a-huaeica</td><td>= [5] + 3.</td></tr> +<tr><td>9.</td><td>a-moacua</td><td>= [5] + 4.</td></tr> +<tr><td>10.</td><td colspan="2">tamoamata (akin to moamati, “hand”).</td></tr> +</table> + +<table class="scale"> +<caption>Aymara.<a id="FNA-316" href="#FN-316"><sup>316</sup></a></caption> + +<tr><td>1.</td><td colspan="2">maya.</td></tr> +<tr><td>2.</td><td colspan="2">paya.</td></tr> +<tr><td>3.</td><td colspan="2">kimsa.</td></tr> +<tr><td>4.</td><td colspan="2">pusi.</td></tr> +<tr><td>5.</td><td colspan="2">piska.</td></tr> +<tr><td>6.</td><td colspan="2">tsokta.</td></tr> +<tr><td>7.</td><td>pa-kalko</td><td>= 2 + 5.</td></tr> +<tr><td>8.</td><td>kimsa-kalko</td><td>= 3 + 5.</td></tr> +<tr><td>9.</td><td>pusi-kalko</td><td>= 4 + 5.</td></tr> +<tr><td>10.</td><td colspan="2">tunka.</td></tr> +</table> + +<table class="scale"> +<caption>Caribs of Essequibo, Guiana.<a id="FNA-317" href="#FN-317"><sup>317</sup></a></caption> + +<tr><td>1.</td><td colspan="2">oween.</td></tr> +<tr><td>2.</td><td colspan="2">oko.</td></tr> +<tr><td>3.</td><td colspan="2">oroowa.</td></tr> +<tr><td>4.</td><td colspan="2">oko-baimema.</td></tr> +<tr><td>5.</td><td>wineetanee</td><td>= 1 hand.</td></tr> +<tr><td>6.</td><td>owee-puimapo</td><td>= 1 again?</td></tr> +<tr><td>7.</td><td>oko-puimapo</td><td>= 2 again?</td></tr> +<tr><td>8.</td><td>oroowa-puimapo</td><td>= 3 again?</td></tr> +<tr><td>9.</td><td>oko-baimema-puimapo</td><td>= 4 again?</td></tr> +<tr><td>10.</td><td colspan="2">oween-abatoro.</td></tr> +</table> + +<table class="scale"> +<caption>Carib.<a id="FNA-318" href="#FN-318"><sup>318</sup></a> (Roucouyenne?)</caption> + +<tr><td>1.</td><td colspan="2">aban, amoin.</td></tr> +<tr><td>2.</td><td colspan="2">biama.</td></tr> +<tr><td>3.</td><td colspan="2">eleoua.</td></tr> +<tr><td>4.</td><td>biam-bouri</td><td>= 2 again?</td></tr> +<tr><td>5.</td><td colspan="2">ouacabo-apourcou-aban-tibateli.</td></tr> +<tr><td>6.</td><td colspan="2">aban laoyagone-ouacabo-apourcou.</td></tr> +<tr><td>7.</td><td colspan="2">biama laoyagone-ouacabo-apourcou.</td></tr> +<tr><td>8.</td><td colspan="2">eleoua laoyagone-ouacabo-apourcou.</td></tr> +<tr><td>9.</td><td colspan="2">——</td></tr> +<tr><td>10.</td><td colspan="2">chon noucabo.</td></tr> +</table> + +<p><a class="pgnum" id="page-167" title="- 167 -"></a>It is unfortunate that the meanings of these remarkable +numerals cannot be given. The counting is evidently +quinary, but the terms used must have been purely +descriptive expressions, having their origin undoubtedly +in certain gestures or finger motions. The numerals +obtained from this region, and from the tribes to the +south and east of the Carib country, are especially rich +in digital terms, and an analysis of the above numerals +would probably show clearly the mental steps through +which this people passed in constructing the rude scale +which served for the expression of their ideas of number.</p> + +<table class="scale"> +<caption>Kiriri.<a id="FNA-319" href="#FN-319"><sup>319</sup></a></caption> + +<tr><td>1.</td><td colspan="2">biche.</td></tr> +<tr><td>2.</td><td colspan="2">watsani.</td></tr> +<tr><td>3.</td><td colspan="2">watsani dikie.</td></tr> +<tr><td>4.</td><td colspan="2">sumara oroba.</td></tr> +<tr><td>5.</td><td>mi biche misa</td><td>= 1 hand.</td></tr> +<tr><td>6.</td><td colspan="2">mirepri bu-biche misa sai.</td></tr> +<tr><td>7.</td><td colspan="2">mirepri watsani misa sai.</td></tr> +<tr><td>8.</td><td colspan="2">mirepri watsandikie misa sai.</td></tr> +<tr><td>9.</td><td colspan="2">mirepri sumara oraba sai.</td></tr> +<tr><td>10.</td><td>mikriba misa sai</td><td>= both hands.</td></tr> +</table> + +<table class="scale"> +<caption>Cayubaba<a id="FNA-320" href="#FN-320"><sup>320</sup></a></caption> +<tr><td>1.</td><td colspan="2">pebi.</td></tr> +<tr><td>2.</td><td colspan="2">mbeta.</td></tr> +<tr><td>3.</td><td colspan="2">kimisa.</td></tr> +<tr><td>4.</td><td colspan="2">pusi.</td></tr> +<tr><td>5.</td><td colspan="2">pisika.</td></tr> +<tr><td>6.</td><td colspan="2">sukuta.</td></tr> +<tr><td>7.</td><td>pa-kaluku</td><td>= 2 again?</td></tr> +<tr><td>8.</td><td>kimisa-kaluku</td><td>= 3 again?</td></tr> +<tr><td>9.</td><td>pusu-kaluku</td><td>= 4 again?</td></tr> +<tr><td>10.</td><td colspan="2">tunka.</td></tr> +</table> + +<table class="scale"> +<caption>Sapibocona<a href="#FN-320"><sup>320</sup></a></caption> + +<tr><td>1.</td><td colspan="2">karata.</td></tr> +<tr><td>2.</td><td colspan="2">mitia.</td></tr> +<tr><td>3.</td><td colspan="2">kurapa.</td></tr> +<tr><td>4.</td><td colspan="2">tsada.</td></tr> +<tr><td>5.</td><td colspan="2">maidara (from <i>arue</i>, hand).</td></tr> +<tr><td>6.</td><td>karata-rirobo</td><td>= 1 hand with.</td></tr> +<tr><td>7.</td><td>mitia-rirobo</td><td>= 2 hand with.</td></tr> +<tr><td>8.</td><td>kurapa-rirobo</td><td>= 3 hand with.</td></tr> +<tr><td>9.</td><td>tsada-rirobo</td><td>= 4 hand with.</td></tr> +<tr><td>10.</td><td>bururutse</td><td>= hand hand.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-168" title="- 168 -"></a>Ticuna.<a id="FNA-321" href="#FN-321"><sup>321</sup></a></caption> + +<tr><td>1.</td><td colspan="2">hueih.</td></tr> +<tr><td>2.</td><td colspan="2">tarepueh.</td></tr> +<tr><td>3.</td><td colspan="2">tomepueh.</td></tr> +<tr><td>4.</td><td colspan="2">aguemoujih</td></tr> +<tr><td>5.</td><td colspan="2">hueamepueh.</td></tr> +<tr><td>6.</td><td>naïmehueapueh</td><td>= 5 + 1.</td></tr> +<tr><td>7.</td><td>naïmehueatareh</td><td>= 5 + 2.</td></tr> +<tr><td>8.</td><td>naïmehueatameapueh</td><td>= 5 + 3.</td></tr> +<tr><td>9.</td><td>gomeapueh</td><td>= 10 − 1.</td></tr> +<tr><td>10.</td><td colspan="2">gomeh.</td></tr> +</table> + +<table class="scale"> +<caption>Yanua.<a id="FNA-322" href="#FN-322"><sup>322</sup></a></caption> + +<tr><td>1.</td><td colspan="2">tckini.</td></tr> +<tr><td>2.</td><td colspan="2">nanojui.</td></tr> +<tr><td>3.</td><td colspan="2">munua.</td></tr> +<tr><td>4.</td><td>naïrojuino</td><td>= 2d 2.</td></tr> +<tr><td>5.</td><td colspan="2">tenaja.</td></tr> +<tr><td>6.</td><td>teki-natea</td><td>= 1 again?</td></tr> +<tr><td>7.</td><td>nanojui-natea</td><td>= 2 again?</td></tr> +<tr><td>8.</td><td>munua-natea</td><td>= 3 again?</td></tr> +<tr><td>9.</td><td>naïrojuino-natea</td><td>= 4 again?</td></tr> +<tr><td>10.</td><td>huijejuino</td><td>= 2 × 5?</td></tr> +</table> + +<p>The foregoing examples will show with considerable +fulness the wide dispersion of the quinary scale. Every +part of the world contributes its share except Europe, +where the only exceptions to the universal use of the +decimal system are the half-dozen languages, which still +linger on its confines, whose number base is the vigesimal. +Not only is there no living European tongue +possessing a quinary number system, but no trace of +this method of counting is found in any of the numerals +of the earlier forms of speech, which have now +become obsolete. The only possible exceptions of which +I can think are the Greek <span class="greek">πεμπάζειν</span>, to count by fives, +and a few kindred words which certainly do hint at a +remote antiquity in which the ancestors of the Greeks +counted on their fingers, and so grouped their units +into fives. The Roman notation, the familiar I., II., III., +IV. (originally IIII.), V., VI., etc., with equal certainty +suggests quinary counting, but the Latin language +<a class="pgnum" id="page-169" title="- 169 -"></a>contains no vestige of anything of the kind, and the +whole range of Latin literature is silent on this point, +though it contains numerous references to finger counting. +It is quite within the bounds of possibility that +the prehistoric nations of Europe possessed and used a +quinary numeration. But of these races the modern +world knows nothing save the few scanty facts that +can be gathered from the stone implements which have +now and then been brought to light. Their languages +have perished as utterly as have the races themselves, +and speculation concerning them is useless. Whatever +their form of numeration may have been, it has left +no perceptible trace on the languages by which they +were succeeded. Even the languages of northern and +central Europe which were contemporary with the +Greek and Latin of classical times have, with the +exception of the Celtic tongues of the extreme North-west, +left behind them but meagre traces for the +modern student to work on. We presume that the +ancient Gauls and Goths, Huns and Scythians, and +other barbarian tribes had the same method of numeration +that their descendants now have; and it is a +matter of certainty that the decimal scale was, at that +time, not used with the universality which now obtains; +but wherever the decimal was not used, the universal +method was vigesimal; and that the quinary ever had +anything of a foothold in Europe is only to be guessed +from its presence to-day in almost all of the other +corners of the world.</p> + +<p><a class="pgnum" id="page-170" title="- 170 -"></a>From the fact that the quinary is that one of the +three natural scales with the smallest base, it has been +conjectured that all tribes possess, at some time in +their history, a quinary numeration, which at a later +period merges into either the decimal or the vigesimal, +and thus disappears or forms with one of the latter a +mixed system.<a id="FNA-323" href="#FN-323"><sup>323</sup></a> In support of this theory it is urged +that extensive regions which now show nothing but +decimal counting were, beyond all reasonable doubt, +quinary. It is well known, for example, that the decimal +system of the Malays has spread over almost the +entire Polynesian region, displacing whatever native +scales it encountered. The same phenomenon has been +observed in Africa, where the Arab traders have disseminated +their own numeral system very widely, the +native tribes adopting it or modifying their own scales +in such a manner that the Arab influence is detected +without difficulty.</p> + +<p>In view of these facts, and of the extreme readiness +with which a tribe would through its finger counting +fall into the use of the quinary method, it does not at +first seem improbable that the quinary was <i>the</i> original +system. But an extended study of the methods of +counting in vogue among the uncivilized races of all +parts of the world has shown that this theory is entirely +untenable. The decimal scale is no less simple +in its structure than the quinary; and the savage, as +<a class="pgnum" id="page-171" title="- 171 -"></a>he extends the limit of his scale from 5 to 6, may call +his new number 5-1, or, with equal probability, give it +an entirely new name, independent in all respects of +any that have preceded it. With the use of this new +name there may be associated the conception of “5 +and 1 more”; but in such multitudes of instances the +words employed show no trace of any such meaning, that +it is impossible for any one to draw, with any degree +of safety, the inference that the signification was originally +there, but that the changes of time had wrought +changes in verbal form so great as to bury it past the +power of recovery. A full discussion of this question +need not be entered upon here. But it will be of interest +to notice two or three numeral scales in which +the quinary influence is so faint as to be hardly discernible. +They are found in considerable numbers +among the North American Indian languages, as may +be seen by consulting the vocabularies that have been +prepared and published during the last half century.<a id="FNA-324" href="#FN-324"><sup>324</sup></a> +From these I have selected the following, which are +sufficient to illustrate the point in question:</p> + +<table class="scale"> +<caption>Quappa.</caption> + +<tr><td>1.</td><td>milchtih.</td></tr> +<tr><td>2.</td><td>nonnepah.</td></tr> +<tr><td>3.</td><td>dahghenih.</td></tr> +<tr><td>4.</td><td>tuah.</td></tr> +<tr><td>5.</td><td>sattou.</td></tr> +<tr><td>6.</td><td>schappeh.</td></tr> +<tr><td>7.</td><td>pennapah.</td></tr> +<tr><td>8.</td><td>pehdaghenih.</td></tr> +<tr><td>9.</td><td>schunkkah.</td></tr> +<tr><td>10.</td><td>gedeh bonah.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-172" title="- 172 -"></a>Terraba.<a id="FNA-325" href="#FN-325"><sup>325</sup></a></caption> + +<tr><td>1.</td><td>krara.</td></tr> +<tr><td>2.</td><td>krowü.</td></tr> +<tr><td>3.</td><td>krom miah.</td></tr> +<tr><td>4.</td><td>krob king.</td></tr> +<tr><td>5.</td><td>krasch kingde.</td></tr> +<tr><td>6.</td><td>terdeh.</td></tr> +<tr><td>7.</td><td>kogodeh.</td></tr> +<tr><td>8.</td><td>kwongdeh.</td></tr> +<tr><td>9.</td><td>schkawdeh.</td></tr> +<tr><td>10.</td><td>dwowdeh.</td></tr> +</table> + +<table class="scale"> +<caption>Mohican</caption> + +<tr><td>1.</td><td>ngwitloh.</td></tr> +<tr><td>2.</td><td>neesoh.</td></tr> +<tr><td>3.</td><td>noghhoh.</td></tr> +<tr><td>4.</td><td>nauwoh.</td></tr> +<tr><td>5.</td><td>nunon.</td></tr> +<tr><td>6.</td><td>ngwittus.</td></tr> +<tr><td>7.</td><td>tupouwus.</td></tr> +<tr><td>8.</td><td>ghusooh.</td></tr> +<tr><td>9.</td><td>nauneeweh.</td></tr> +<tr><td>10.</td><td>mtannit.</td></tr> +</table> + +<p>In the Quappa scale 7 and 8 appear to be derived +from 2 and 3, while 6 and 9 show no visible trace +of kinship with 1 and 4. In Mohican, on the other +hand, 6 and 9 seem to be derived from 1 and 4, while +7 and 8 have little or no claim to relationship with +2 and 3. In some scales a single word only is found +in the second quinate to indicate that 5 was originally +the base on which the system rested. It is hardly to +be doubted, even, that change might affect each and +every one of the numerals from 5 to 10 or 6 to 9, so +that a dependence which might once have been easily +detected is now unrecognizable.</p> + +<p>But if this is so, the natural and inevitable question +follows—might not this have been the history of all +numeral scales now purely decimal? May not the +changes of time have altered the compounds which +were once a clear indication of quinary counting, until +<a class="pgnum" id="page-173" title="- 173 -"></a>no trace remains by which they can be followed back +to their true origin? Perhaps so. It is not in the +least degree probable, but its possibility may, of course, +be admitted. But even then the universality of quinary +counting for primitive peoples is by no means +established. In Chapter II, examples were given of races +which had no number base. Later on it was observed +that in Australia and South America many tribes used +2 as their number base; in some cases counting on past +5 without showing any tendency to use that as a new +unit. Again, through the habit of counting upon the +finger joints, instead of the fingers themselves, the use +of 3 as a base is brought into prominence, and 6 and +9 become 2 threes and 3 threes, respectively, instead of +5 + 1 and 5 + 4. The same may be noticed of 4. Counting +by means of his fingers, without including the +thumbs, the savage begins by dividing into fours instead +of fives. Traces of this form of counting are somewhat +numerous, especially among the North American aboriginal +tribes. Hence the quinary form of counting, +however widespread its use may be shown to be, can +in no way be claimed as the universal method of any +stage of development in the history of mankind.</p> + +<p>In the vast majority of cases, the passage from the +base to the next succeeding number in any scale, is +clearly defined. But among races whose intelligence is +of a low order, or—if it be permissible to express +it in this way—among races whose number sense is +feeble, progression from one number to the next is not +<a class="pgnum" id="page-174" title="- 174 -"></a>always in accordance with any well-defined law. After +one or two distinct numerals the count may, as in the +case of the Veddas and the Andamans, proceed by finger +pantomime and by the repetition of the same word. +Occasionally the same word is used for two successive +numbers, some gesture undoubtedly serving to distinguish +the one from the other in the savage's mind. +Examples of this are not infrequent among the forest +tribes of South America. In the Tariana dialect 9 +and 10 are expressed by the same word, <i>paihipawalianuda;</i> +in Cobeu, 8 and 9 by <i>pepelicoloblicouilini;</i> in +Barre, 4, 5, and 9 by <i>ualibucubi.</i><a id="FNA-326" href="#FN-326"><sup>326</sup></a> In other languages the +change from one numeral to the next is so slight that +one instinctively concludes that the savage is forming +in his own mind another, to him new, numeral immediately +from the last. In such cases the entire number +system is scanty, and the creeping hesitancy with which +progress is made is visible in the forms which the numerals +are made to take. A single illustration or two +of this must suffice; but the ones chosen are not isolated +cases. The scale of the Macunis,<a id="FNA-327" href="#FN-327"><sup>327</sup></a> one of the numerous +tribes of Brazil, is</p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">pocchaenang.</td></tr> +<tr><td>2.</td><td colspan="2">haihg.</td></tr> +<tr><td>3.</td><td colspan="2">haigunhgnill.</td></tr> +<tr><td>4.</td><td colspan="2">haihgtschating.</td></tr> +<tr><td>5.</td><td>haihgtschihating</td><td>= another 4?</td></tr> +<tr><td>6.</td><td>hathig-stchihathing</td><td>= 2-4?</td></tr> +<tr><td>7.</td><td>hathink-tschihathing</td><td>= 2-5?</td></tr> +<tr><td>8.</td><td>hathink-tschihating</td><td>= 2 × 4?</td></tr> +</table> + +<p><a class="pgnum" id="page-175" title="- 175 -"></a>The complete absence of—one is tempted to say—any +rhyme or reason from this scale is more than +enough to refute any argument which might tend to +show that the quinary, or any other scale, was ever the +sole number scale of primitive man. Irregular as this is, +the system of the Montagnais fully matches it, as the +subjoined numerals show:<a id="FNA-328" href="#FN-328"><sup>328</sup></a></p> + +<table class="scale"> +<tr><td>1.</td><td colspan="2">inl'are.</td></tr> +<tr><td>2.</td><td colspan="2">nak'e.</td></tr> +<tr><td>3.</td><td colspan="2">t'are.</td></tr> +<tr><td>4.</td><td colspan="2">dinri.</td></tr> +<tr><td>5.</td><td colspan="2">se-sunlare.</td></tr> +<tr><td>6.</td><td>elkke-t'are</td><td>= 2 × 3.</td></tr> +<tr><td>7.</td><td>t'a-ye-oyertan</td><td>= 10 − 3,</td></tr> +<tr><td></td><td class="alt">or inl'as dinri</td><td>= 4 + 3?</td></tr> +<tr><td>8.</td><td>elkke-dinri</td><td>= 2 × 4.</td></tr> +<tr><td>9.</td><td>inl'a-ye-oyertan</td><td>= 10 − 1.</td></tr> +<tr><td>10.</td><td colspan="2">onernan.</td></tr> +</table> + + + + +<h2><a class="pgnum" id="page-176" title="- 176 -"></a>Chapter VII.</h2> +<h3>The Vigesimal System.</h3> + + +<p><span class="initial">In</span> its ordinary development the quinary system is +almost sure to merge into either the decimal or the +vigesimal system, and to form, with one or the other +or both of these, a mixed system of counting. In +Africa, Oceanica, and parts of North America, the +union is almost always with the decimal scale; while +in other parts of the world the quinary and the vigesimal +systems have shown a decided affinity for each +other. It is not to be understood that any geographical +law of distribution has ever been observed which +governs this, but merely that certain families of races +have shown a preference for the one or the other +method of counting. These families, disseminating +their characteristics through their various branches, +have produced certain groups of races which exhibit +a well-marked tendency, here toward the decimal, and +there toward the vigesimal form of numeration. As +far as can be ascertained, the choice of the one or the +other scale is determined by no external circumstances, +but depends solely on the mental characteristics of +the tribes themselves. Environment does not exert any +appreciable influence either. Both decimal and vigesimal +<a class="pgnum" id="page-177" title="- 177 -"></a>numeration are found indifferently in warm and in +cold countries; in fruitful and in barren lands; in +maritime and in inland regions; and among highly +civilized or deeply degraded peoples.</p> + +<p>Whether or not the principal number base of any +tribe is to be 20 seems to depend entirely upon a single +consideration; are the fingers alone used as an aid +to counting, or are both fingers and toes used? If +only the fingers are employed, the resulting scale must +become decimal if sufficiently extended. If use is made +of the toes in addition to the fingers, the outcome must +inevitably be a vigesimal system. Subordinate to either +one of these the quinary may and often does appear. +It is never the principal base in any extended system.</p> + +<p>To the statement just made respecting the origin of +vigesimal counting, exception may, of course, be taken. +In the case of numeral scales like the Welsh, the Nahuatl, +and many others where the exact meanings of the +numerals cannot be ascertained, no proof exists that +the ancestors of these peoples ever used either finger or +toe counting; and the sweeping statement that any +vigesimal scale is the outgrowth of the use of these +natural counters is not susceptible of proof. But so +many examples are met with in which the origin is +clearly of this nature, that no hesitation is felt in putting +the above forward as a general explanation for the +existence of this kind of counting. Any other origin +is difficult to reconcile with observed facts, and still +more difficult to reconcile with any rational theory of +<a class="pgnum" id="page-178" title="- 178 -"></a>number system development. Dismissing from consideration +the quinary scale, let us briefly examine once +more the natural process of evolution through which +the decimal and the vigesimal scales come into being. +After the completion of one count of the fingers the +savage announces his result in some form which definitely +states to his mind the fact that the end of a well-marked +series has been reached. Beginning again, he +now repeats his count of 10, either on his own fingers +or on the fingers of another. With the completion of +the second 10 the result is announced, not in a new +unit, but by means of a duplication of the term already +used. It is scarcely credible that the unit unconsciously +adopted at the termination of the first count +should now be dropped, and a new one substituted in +its place. When the method here described is employed, +20 is not a natural unit to which higher numbers +may be referred. It is wholly artificial; and it +would be most surprising if it were adopted. But if +the count of the second 10 is made on the toes in +place of the fingers, the element of repetition which +entered into the previous method is now wanting. Instead +of referring each new number to the 10 already +completed, the savage is still feeling his way along, +designating his new terms by such phrases as “1 on +the foot,” “2 on the other foot,” etc. And now, when +20 is reached, a single series is finished instead of a +double series as before; and the result is expressed in +one of the many methods already noticed—“one man,” +<a class="pgnum" id="page-179" title="- 179 -"></a>“hands and feet,” “the feet finished,” “all the fingers +of hands and feet,” or some equivalent formula. Ten +is no longer the natural base. The number from which +the new start is made is 20, and the resulting scale is +inevitably vigesimal. If pebbles or sticks are used +instead of fingers, the system will probably be decimal. +But back of the stick and pebble counting the 10 natural +counters always exist, and to them we must always +look for the origin of this scale.</p> + +<p>In any collection of the principal vigesimal number +systems of the world, one would naturally begin with +those possessed by the Celtic races of Europe. These +races, the earliest European peoples of whom we have +any exact knowledge, show a preference for counting +by twenties, which is almost as decided as that manifested +by Teutonic races for counting by tens. It has +been conjectured by some writers that the explanation +for this was to be found in the ancient commercial +intercourse which existed between the Britons and the +Carthaginians and Phœnicians, whose number systems +showed traces of a vigesimal tendency. Considering +the fact that the use of vigesimal counting was universal +among Celtic races, this explanation is quite +gratuitous. The reason why the Celts used this method +is entirely unknown, and need not concern investigators +in the least. But the fact that they did use it +is important, and commands attention. The five Celtic +languages, Breton, Irish, Welsh, Manx, and Gaelic, contain +the following well-defined vigesimal scales. Only +<a class="pgnum" id="page-180" title="- 180 -"></a>the principal or characteristic numerals are given, those +being sufficient to enable the reader to follow intelligently +the growth of the systems. Each contains the +decimal element also, and is, therefore, to be regarded +as a mixed decimal-vigesimal system.</p> + +<table class="scale"> +<caption>Irish.<a id="FNA-329" href="#FN-329"><sup>329</sup></a></caption> + +<tr><td>10.</td><td colspan="2">deic.</td></tr> +<tr><td>20.</td><td colspan="2">fice.</td></tr> +<tr><td>30.</td><td>triocad</td><td>= 3-10</td></tr> +<tr><td>40.</td><td>da ficid</td><td>= 2-20.</td></tr> +<tr><td>50.</td><td>caogad</td><td>= 5-10.</td></tr> +<tr><td>60.</td><td>tri ficid</td><td>= 3-20.</td></tr> +<tr><td>70.</td><td>reactmoga</td><td>= 7-10.</td></tr> +<tr><td>80.</td><td>ceitqe ficid</td><td>= 4-20.</td></tr> +<tr><td>90.</td><td>nocad</td><td>= 9-10.</td></tr> +<tr><td>100.</td><td colspan="2">cead.</td></tr> +<tr><td>1000.</td><td colspan="2">mile.</td></tr> +</table> + +<table class="scale"> +<caption>Gaelic.<a id="FNA-330" href="#FN-330"><sup>330</sup></a></caption> + +<tr><td>10.</td><td colspan="2">deich.</td></tr> +<tr><td>20.</td><td colspan="2">fichead.</td></tr> +<tr><td>30.</td><td>deich ar fichead</td><td>= 10 + 20.</td></tr> +<tr><td>40.</td><td>da fhichead</td><td>= 2-20.</td></tr> +<tr><td>50.</td><td>da fhichead is deich</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>tri fichead</td><td>= 3-20.</td></tr> +<tr><td>70.</td><td>tri fichead is deich</td><td>= 60 + 10.</td></tr> +<tr><td>80.</td><td>ceithir fichead</td><td>= 4-20.</td></tr> +<tr><td>90.</td><td>ceithir fichead is deich</td><td>= 80 + 10.</td></tr> +<tr><td>100.</td><td colspan="2">ceud.</td></tr> +<tr><td>1000.</td><td colspan="2">mile.</td></tr> +</table> + + +<table class="scale"> +<caption>Welsh.<a id="FNA-331" href="#FN-331"><sup>331</sup></a></caption> + +<tr><td>10.</td><td colspan="2">deg.</td></tr> +<tr><td>20.</td><td colspan="2">ugain.</td></tr> +<tr><td>30.</td><td>deg ar hugain</td><td>= 10 + 20.</td></tr> +<tr><td>40.</td><td>deugain</td><td>= 2-20.</td></tr> +<tr><td>50.</td><td>deg a deugain</td><td>= 10 + 40.</td></tr> +<tr><td>60.</td><td>trigain</td><td>= 3-20.</td></tr> +<tr><td>70.</td><td>deg a thrigain</td><td>= 10 + 60.</td></tr> +<tr><td>80.</td><td>pedwar ugain</td><td>= 4-20.</td></tr> +<tr><td>90.</td><td>deg a pedwar ugain</td><td>= 80 + 10.</td></tr> +<tr><td>100.</td><td colspan="2">cant.</td></tr> +</table> + +<table class="scale"> +<caption>Manx.<a id="FNA-332" href="#FN-332"><sup>332</sup></a></caption> + +<tr><td>10.</td><td colspan="2">jeih.</td></tr> +<tr><td>20.</td><td colspan="2">feed.</td></tr> +<tr><td>30.</td><td>yn jeih as feed</td><td>= 10 + 20.</td></tr> +<tr><td>40.</td><td>daeed</td><td>= 2-20.</td></tr> +<tr><td>50.</td><td>jeih as daeed</td><td>= 10 + 40.</td></tr> +<tr><td>60.</td><td>three-feed</td><td>= 3-20.</td></tr> +<tr><td>70.</td><td>three-feed as jeih</td><td>= 60 + 10.</td></tr> +<tr><td>80.</td><td>kiare-feed</td><td>= 4-20.</td></tr> +<tr><td>100.</td><td colspan="2">keead.</td></tr> +<tr><td>1000.</td><td colspan="2">thousane, or jeih cheead.</td></tr> +</table> + +<table class="scale"> +<caption><a class="pgnum" id="page-181" title="- 181 -"></a>Breton.<a id="FNA-333" href="#FN-333"><sup>333</sup></a></caption> + +<tr><td>10.</td><td colspan="2">dec.</td></tr> +<tr><td>20.</td><td colspan="2">ueguend.</td></tr> +<tr><td>30.</td><td>tregond</td><td>= 3-10.</td></tr> +<tr><td>40.</td><td>deu ueguend</td><td>= 2-20.</td></tr> +<tr><td>50.</td><td>hanter hand</td><td>= half hundred.</td></tr> +<tr><td>60.</td><td>tri ueguend</td><td>= 3-20.</td></tr> +<tr><td>70.</td><td>dec ha tri ueguend</td><td>= 10 + 60.</td></tr> +<tr><td>80.</td><td>piar ueguend</td><td>= 4-20.</td></tr> +<tr><td>90.</td><td>dec ha piar ueguend</td><td>= 10 + 80.</td></tr> +<tr><td>100.</td><td colspan="2">cand.</td></tr> +<tr><td>120.</td><td>hueh ueguend</td><td>= 6-20.</td></tr> +<tr><td>140.</td><td>seih ueguend</td><td>= 7-20.</td></tr> +<tr><td>160.</td><td>eih ueguend</td><td>= 8-20.</td></tr> +<tr><td>180.</td><td>nau ueguend</td><td>= 9-20.</td></tr> +<tr><td>200.</td><td>deu gand</td><td>= 2-100.</td></tr> +<tr><td>240.</td><td>deuzec ueguend</td><td>= 12-20.</td></tr> +<tr><td>280.</td><td>piarzec ueguend</td><td>= 14-20.</td></tr> +<tr><td>300.</td><td colspan="2">tri hand, or pembzec ueguend.</td></tr> +<tr><td>400.</td><td>piar hand</td><td>= 4-100.</td></tr> +<tr><td>1000.</td><td colspan="2">mil.</td></tr> +</table> + +<p>These lists show that the native development of +the Celtic number systems, originally showing a strong +preference for the vigesimal method of progression, has +been greatly modified by intercourse with Teutonic +and Latin races. The higher numerals in all these +languages, and in Irish many of the lower also, are +seen at a glance to be decimal. Among the scales here +given the Breton, the legitimate descendant of the +ancient Gallic, is especially interesting; but here, just +as in the other Celtic tongues, when we reach 1000, +the familiar Latin term for that number appears in the +various corruptions of <i>mille</i>, 1000, which was carried +into the Celtic countries by missionary and military +influences.</p> + +<p>In connection with the Celtic language, mention +must be made of the persistent vigesimal element +which has held its place in French. The ancient +<a class="pgnum" id="page-182" title="- 182 -"></a>Gauls, while adopting the language of their conquerors, +so far modified the decimal system of Latin as to +replace the natural <i>septante</i>, 70, <i>octante</i>, 80, <i>nonante</i>, +90, by <i>soixante-dix</i>, 60-10, +<i>quatre-vingt</i>, 4-20, and <i>quatrevingt-dix</i>, +4-20-10. From 61 to 99 the French method +of counting is wholly vigesimal, except for the presence +of the one word <i>soixante</i>. In old French this element +was still more pronounced. <i>Soixante</i> had not +yet appeared; and 60 and 70 were <i>treis vinz</i>, +3-20, and +<i>treis vinz et dis</i>, 3-20 and 10 respectively. Also, 120 +was <i>six vinz</i>, 6-20, 140 was <i>sept-vinz</i>, etc.<a id="FNA-334" href="#FN-334"><sup>334</sup></a> How far +this method ever extended in the French language +proper, it is, perhaps, impossible to say; but from the +name of an almshouse, <i>les quinze-vingts</i>,<a id="FNA-335" href="#FN-335"><sup>335</sup></a> which formerly +existed in Paris, and was designed as a home for 300 +blind persons, and from the <i>pembzek-ueguent</i>, +15-20, of +the Breton, which still survives, we may infer that it +was far enough to make it the current system of +common life.</p> + +<p>Europe yields one other example of vigesimal counting, +in the number system of the Basques. Like most +of the Celtic scales, the Basque seems to become decimal +above 100. It does not appear to be related to +any other European system, but to be quite isolated +philologically. The higher units, as <i>mila</i>, 1000, are +probably borrowed, and not native. The tens in the +Basque scale are:<a id="FNA-336" href="#FN-336"><sup>336</sup></a></p> + +<table class="scale"> +<tr><td><a class="pgnum" id="page-183" title="- 183 -"></a>10.</td><td colspan="2">hamar.</td></tr> +<tr><td>20.</td><td colspan="2">hogei.</td></tr> +<tr><td>30.</td><td>hogei eta hamar</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>berrogei</td><td>= 2-20.</td></tr> +<tr><td>50.</td><td>berrogei eta hamar</td><td>= 2-20 + 10.</td></tr> +<tr><td>60.</td><td>hirurogei</td><td>= 3-20.</td></tr> +<tr><td>70.</td><td>hirurogei eta hamar</td><td>= 3-20 + 10.</td></tr> +<tr><td>80.</td><td>laurogei</td><td>= 4-20.</td></tr> +<tr><td>90.</td><td>laurogei eta hamar</td><td>= 4-20 + 10.</td></tr> +<tr><td>100.</td><td colspan="2">ehun.</td></tr> +<tr><td>1000.</td><td colspan="2"><i>milla</i>.</td></tr> +</table> + +<p>Besides these we find two or three numeral scales in +Europe which contain distinct traces of vigesimal counting, +though the scales are, as a whole, decidedly decimal. +The Danish, one of the essentially Germanic +languages, contains the following numerals:</p> + +<table class="scale"> +<tr><td>30.</td><td>tredive</td><td>= 3-10.</td></tr> +<tr><td>40.</td><td>fyrretyve</td><td>= 4-10.</td></tr> +<tr><td>50.</td><td>halvtredsindstyve</td><td>= half (of 20) from 3-20.</td></tr> +<tr><td>60.</td><td>tresindstyve</td><td>= 3-20.</td></tr> +<tr><td>70.</td><td>halvfierdsindstyve</td><td>= half from 4-20.</td></tr> +<tr><td>80.</td><td>fiirsindstyve</td><td>= 4-20.</td></tr> +<tr><td>90.</td><td>halvfemsindstyve</td><td>= half from 5-20.</td></tr> +<tr><td>100.</td><td colspan="2">hundrede.</td></tr> +</table> + +<p>Germanic number systems are, as a rule, pure decimal +systems; and the Danish exception is quite remarkable. +We have, to be sure, such expressions in English as +<i>three score</i>, <i>four score</i>, etc., and the Swedish, Icelandic, +and other languages of this group have similar terms. +Still, these are not pure numerals, but auxiliary words +rather, which belong to the same category as <i>pair</i>, +<i>dozen</i>, <i>dizaine</i>, etc., while the Danish words just given +are the ordinary numerals which form a part of the +every-day vocabulary of that language. The method +by which this scale expresses 50, 70, and 90 is especially +<a class="pgnum" id="page-184" title="- 184 -"></a>noticeable. It will be met with again, and +further examples of its occurrence given.</p> + +<p>In Albania there exists one single fragment of vigesimal +numeration, which is probably an accidental compound +rather than the remnant of a former vigesimal +number system. With this single exception the Albanian +scale is of regular decimal formation. A few of +the numerals are given for the sake of comparison:<a id="FNA-337" href="#FN-337"><sup>337</sup></a></p> + +<table class="scale"> +<tr><td>30.</td><td>tridgiete</td><td>= 3-10.</td></tr> +<tr><td>40.</td><td>dizet</td><td>= 2-20.</td></tr> +<tr><td>50.</td><td>pesedgiete</td><td>= 5-10.</td></tr> +<tr><td>60.</td><td>giastedgiete</td><td>= 6-10, etc.</td></tr> +</table> + +<p>Among the almost countless dialects of Africa we find +a comparatively small number of vigesimal number systems. +The powers of the negro tribes are not strongly +developed in counting, and wherever their numeral scales +have been taken down by explorers they have almost +always been found to be decimal or quinary-decimal. +The small number I have been able to collect are here +given. They are somewhat fragmentary, but are as +complete as it was possible to make them.</p> + +<table class="scale"> +<caption>Affadeh.<a id="FNA-338" href="#FN-338"><sup>338</sup></a></caption> +<tr><td>10.</td><td colspan="2">dekang.</td></tr> +<tr><td>20.</td><td colspan="2">degumm.</td></tr> +<tr><td>30.</td><td colspan="2">piaske.</td></tr> +<tr><td>40.</td><td>tikkumgassih</td><td>= 20 × 2.</td></tr> +<tr><td>50.</td><td>tikkumgassigokang</td><td>= 20 × 2 + 10.</td></tr> +<tr><td>60.</td><td>tikkumgakro</td><td>= 20 × 3.</td></tr> +<tr><td>70.</td><td>dungokrogokang</td><td>= 20 × 3 + 10.</td></tr> +<tr><td>80.</td><td>dukumgade</td><td>= 20 × 4.</td></tr> +<tr><td>90.</td><td>dukumgadegokang</td><td>= 20 × 4 + 10.</td></tr> +<tr><td>100.</td><td colspan="2">miah (borrowed from the Arabs).</td></tr> +</table> + + +<table class="scale"> +<caption><a class="pgnum" id="page-185" title="- 185 -"></a>Ibo.<a id="FNA-339" href="#FN-339"><sup>339</sup></a></caption> + +<tr><td>10.</td><td colspan="2">iri.</td></tr> +<tr><td>20.</td><td colspan="2">ogu.</td></tr> +<tr><td>30.</td><td>ogu n-iri</td><td>= 20 + 10,</td></tr> +<tr><td></td><td class="alt">or iri ato</td><td>= 10 × 3.</td></tr> +<tr><td>40.</td><td>ogu abuo</td><td>= 20 × 2,</td></tr> +<tr><td></td><td class="alt">or iri anno</td><td>= 10 × 4.</td></tr> +<tr><td>100.</td><td>ogu ise</td><td>= 20 × 5.</td></tr> +</table> + +<table class="scale"> +<caption>Vei.<a id="FNA-340" href="#FN-340"><sup>340</sup></a></caption> + +<tr><td>10.</td><td colspan="2">tan.</td></tr> +<tr><td>20.</td><td>mo bande</td><td>= a person finished.</td></tr> +<tr><td>30.</td><td>mo bande ako tan</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>mo fera bande</td><td>= 2 × 20.</td></tr> +<tr><td>100.</td><td>mo soru bande</td><td>= 5 persons finished.</td></tr> +</table> + +<table class="scale"> +<caption>Yoruba.<a id="FNA-341" href="#FN-341"><sup>341</sup></a></caption> + +<tr><td>10.</td><td colspan="2">duup.</td></tr> +<tr><td>20.</td><td colspan="2">ogu.</td></tr> +<tr><td>30.</td><td colspan="2">ogbo.</td></tr> +<tr><td>40.</td><td>ogo-dzi</td><td>= 20 × 2.</td></tr> +<tr><td>60.</td><td>ogo-ta</td><td>= 20 × 3.</td></tr> +<tr><td>80.</td><td>ogo-ri</td><td>= 20 × 4.</td></tr> +<tr><td>100.</td><td>ogo-ru</td><td>= 20 × 5.</td></tr> +<tr><td>120.</td><td>ogo-fa</td><td>= 20 × 6.</td></tr> +<tr><td>140.</td><td>ogo-dze</td><td>= 20 × 7.</td></tr> +<tr><td>160.</td><td>ogo-dzo</td><td>= 20 × 8, etc.</td></tr> +</table> + +<table class="scale"> +<caption>Efik.<a id="FNA-342" href="#FN-342"><sup>342</sup></a></caption> + +<tr><td>10.</td><td colspan="2">duup.</td></tr> +<tr><td>20.</td><td colspan="2">edip.</td></tr> +<tr><td>30.</td><td>edip-ye-duup</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>aba</td><td>= 20 × 2.</td></tr> +<tr><td>60.</td><td>ata</td><td>= 20 × 3.</td></tr> +<tr><td>80.</td><td>anan</td><td>= 20 × 4.</td></tr> +<tr><td>100.</td><td colspan="2">ikie.</td></tr> +</table> + +<p>The Yoruba scale, to which reference has already been +made, <a href="#page-70">p. 70</a>, again shows its peculiar structure, by continuing +its vigesimal formation past 100 with no interruption +in its method of numeral building. It will be +remembered that none of the European scales showed +this persistency, but passed at that point into decimal +numeration. This will often be found to be the case; +but now and then a scale will come to our notice whose +vigesimal structure is continued, without any break, on +into the hundreds and sometimes into the thousands.</p> + + +<table class="scale"> +<caption><a class="pgnum" id="page-186" title="- 186 -"></a>Bongo.<a id="FNA-343" href="#FN-343"><sup>343</sup></a></caption> + +<tr><td>10.</td><td colspan="2">kih.</td></tr> +<tr><td>20.</td><td>mbaba kotu</td><td>= 20 × 1.</td></tr> +<tr><td>40.</td><td>mbaba gnorr</td><td>= 20 × 2.</td></tr> +<tr><td>100.</td><td>mbaba mui</td><td>= 20 × 5.</td></tr> +</table> + +<table class="scale"> +<caption>Mende.<a id="FNA-344" href="#FN-344"><sup>344</sup></a></caption> + +<tr><td>10.</td><td colspan="2">pu.</td></tr> +<tr><td>20.</td><td>nu yela gboyongo mai</td><td>= a man finished.</td></tr> +<tr><td>30.</td><td>nu yela gboyongo mahu pu</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>nu fele gboyongo</td><td>= 2 men finished.</td></tr> +<tr><td>100.</td><td>nu lolu gboyongo</td><td>= 5 men finished.</td></tr> +</table> + +<table class="scale"> +<caption>Nupe.<a id="FNA-345" href="#FN-345"><sup>345</sup></a></caption> + +<tr><td>10.</td><td colspan="2">gu-wo.</td></tr> +<tr><td>20.</td><td colspan="2">esin.</td></tr> +<tr><td>30.</td><td colspan="2">gbonwo.</td></tr> +<tr><td>40.</td><td>si-ba</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td colspan="2">arota.</td></tr> +<tr><td>60.</td><td>sita</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td colspan="2">adoni.</td></tr> +<tr><td>80.</td><td>sini</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td>sini be-guwo</td><td>= 80 + 10.</td></tr> +<tr><td>100.</td><td>sisun</td><td>= 5 × 20.</td></tr> +</table> + +<table class="scale"> +<caption>Logone.<a id="FNA-346" href="#FN-346"><sup>346</sup></a></caption> + +<tr><td>10.</td><td colspan="2">chkan.</td></tr> +<tr><td>20.</td><td colspan="2">tkam.</td></tr> +<tr><td>30.</td><td>tkam ka chkan</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>tkam ksde</td><td>= 20 × 2.</td></tr> +<tr><td>50.</td><td>tkam ksde ka chkan</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>tkam gachkir</td><td>= 20 × 3.</td></tr> +<tr><td>100.</td><td colspan="2">mia (from Arabic).</td></tr> +<tr><td>1000.</td><td colspan="2">debu.</td></tr> +</table> + +<table class="scale"> +<caption>Mundo.<a id="FNA-347" href="#FN-347"><sup>347</sup></a></caption> + +<tr><td>10.</td><td colspan="2">nujorquoi.</td></tr> +<tr><td>20.</td><td colspan="2">tiki bere.</td></tr> +<tr><td>30.</td><td>tiki bire nujorquoi</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>tiki borsa</td><td>= 20 × 2.</td></tr> +<tr><td>50.</td><td>tike borsa nujorquoi</td><td>= 40 + 10.</td></tr> +</table> + +<table class="scale"> +<caption>Mandingo.<a id="FNA-348" href="#FN-348"><sup>348</sup></a></caption> + +<tr><td>10.</td><td colspan="2">tang.</td></tr> +<tr><td>20.</td><td colspan="2">mulu.</td></tr> +<tr><td>30.</td><td>mulu nintang</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>mulu foola</td><td>= 20 × 2.</td></tr> +<tr><td>50.</td><td>mulu foola nintang</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>mulu sabba</td><td>= 20 × 3.</td></tr> +<tr><td>70.</td><td>mulu sabba nintang</td><td>= 60 + 10.</td></tr> +<tr><td>80.</td><td>mulu nani</td><td>= 20 × 4.</td></tr> +<tr><td>90.</td><td>mulu nani nintang</td><td>= 80 + 10.</td></tr> +<tr><td>100.</td><td colspan="2">kemi.</td></tr> +</table> + + +<p><a class="pgnum" id="page-187" title="- 187 -"></a>This completes the scanty list of African vigesimal +number systems that a patient and somewhat extended +search has yielded. It is remarkable that the number is +no greater. Quinary counting is not uncommon in the +“Dark Continent,” and there is no apparent reason why +vigesimal reckoning should be any less common than +quinary. Any one investigating African modes of counting +with the material at present accessible, will find +himself hampered by the fact that few explorers have +collected any except the first ten numerals. This leaves +the formation of higher terms entirely unknown, and +shows nothing beyond the quinary or non-quinary character +of the system. Still, among those which Stanley, +Schweinfurth, Salt, and others have collected, by far the +greatest number are decimal. As our knowledge of +African languages is extended, new examples of the +vigesimal method may be brought to light. But our +present information leads us to believe that they will +be few in number.</p> + +<p>In Asia the vigesimal system is to be found with +greater frequency than in Europe or Africa, but it is +still the exception. As Asiatic languages are much +better known than African, it is probable that the future +will add but little to our stock of knowledge on this +point. New instances of counting by twenties may still +be found in northern Siberia, where much ethnological +work yet remains to be done, and where a tendency +toward this form of numeration has been observed to +exist. But the total number of Asiatic vigesimal scales +<a class="pgnum" id="page-188" title="- 188 -"></a>must always remain small—quite insignificant in comparison +with those of decimal formation.</p> + +<p>In the Caucasus region a group of languages is found, +in which all but three or four contain vigesimal systems. +These systems are as follows:</p> + +<table class="scale"> +<caption>Abkhasia.<a id="FNA-349" href="#FN-349"><sup>349</sup></a></caption> + +<tr><td>10.</td><td colspan="2">zpha-ba.</td></tr> +<tr><td>20.</td><td>gphozpha</td><td>= 2 × 10.</td></tr> +<tr><td>30.</td><td>gphozphei zphaba</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>gphin-gphozpha</td><td>= 2 × 20.</td></tr> +<tr><td>60.</td><td>chin-gphozpha</td><td>= 3 × 20.</td></tr> +<tr><td>80.</td><td>phsin-gphozpha</td><td>= 4 × 20.</td></tr> +<tr><td>100.</td><td colspan="2">sphki.</td></tr> +</table> + +<table class="scale"> +<caption>Avari</caption> + +<tr><td>10.</td><td colspan="2">antsh-go.</td></tr> +<tr><td>20.</td><td colspan="2">qo-go.</td></tr> +<tr><td>30.</td><td colspan="2">lebergo.</td></tr> +<tr><td>40.</td><td>khi-qogo</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>khiqojalda antshgo</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>lab-qogo</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>labqojalda antshgo</td><td>= 60 + 10.</td></tr> +<tr><td>80.</td><td>un-qogo</td><td>= 4 × 20.</td></tr> +<tr><td>100.</td><td colspan="2">nusgo.</td></tr> +</table> + +<table class="scale"> +<caption>Kuri</caption> + +<tr><td>10.</td><td colspan="2">tshud.</td></tr> +<tr><td>20.</td><td colspan="2">chad.</td></tr> +<tr><td>30.</td><td>channi tshud</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td colspan="2">jachtshur.</td></tr> +<tr><td>50.</td><td>jachtshurni tshud</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>put chad</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>putchanni tshud</td><td>= 60 + 10.</td></tr> +<tr><td>80.</td><td>kud-chad</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td>kudchanni tshud</td><td>= 80 + 10.</td></tr> +<tr><td>100.</td><td colspan="2">wis.</td></tr> +</table> + +<table class="scale"> +<caption>Udi</caption> + +<tr><td>10.</td><td colspan="2">witsh.</td></tr> +<tr><td>20.</td><td colspan="2">qa.</td></tr> +<tr><td>30.</td><td>sa-qo-witsh</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>pha-qo</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>pha-qo-witsh</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>chib-qo</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>chib-qo-witsh</td><td>= 60 + 10.</td></tr> +<tr><td>80.</td><td>bip-qo</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td>bip-qo-witsh</td><td>= 80 + 10.</td></tr> +<tr><td>100.</td><td colspan="2">bats.</td></tr> +<tr><td>1000.</td><td colspan="2">hazar (Persian).</td></tr> +</table> + +<table class="scale"> +<caption>Tchetchnia</caption> + +<tr><td>10.</td><td colspan="2">ith.</td></tr> +<tr><td>20.</td><td colspan="2">tqa.</td></tr> +<tr><td>30.</td><td>tqe ith</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>sauz-tqa</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>sauz-tqe ith</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>chuz-tqa</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>chuz-tqe ith</td><td>= 60 + 10.</td></tr> +<tr><td>80.</td><td>w-iez-tqa</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td>w-iez-tqe ith</td><td>= 80 + 10.</td></tr> +<tr><td>100.</td><td colspan="2">b'e.</td></tr> +<tr><td>1000.</td><td colspan="2">ezir (akin to Persian).</td></tr> +</table> + + +<table class="scale"> +<caption><a class="pgnum" id="page-189" title="- 189 -"></a>Thusch</caption> + +<tr><td>10.</td><td colspan="2">itt.</td></tr> +<tr><td>20.</td><td colspan="2">tqa.</td></tr> +<tr><td>30.</td><td>tqa-itt</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>sauz-tq</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>sauz-tqa-itt</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>chouz-tq</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>chouz-tqa-itt</td><td>= 60 + 10.</td></tr> +<tr><td>80.</td><td>dhewuz-tq</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td>dhewuz-tqa-itt</td><td>= 80 + 10.</td></tr> +<tr><td>100.</td><td>phchauz-tq</td><td>= 5 × 20.</td></tr> +<tr><td>200.</td><td>itsha-tq</td><td>= 10 × 20.</td></tr> +<tr><td>300.</td><td>phehiitsha-tq</td><td>= 15 × 20.</td></tr> +<tr><td>1000.</td><td>satsh tqauz-tqa itshatqa</td><td>= 2 × 20 × 20 + 200.</td></tr> +</table> + +<table class="scale"> +<caption>Georgia</caption> + +<tr><td>10.</td><td colspan="2">athi.</td></tr> +<tr><td>20.</td><td colspan="2">otsi.</td></tr> +<tr><td>30.</td><td>ots da athi</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>or-m-otsi</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>ormots da athi</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>sam-otsi</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>samots da athi</td><td>= 60 + 10.</td></tr> +<tr><td>80.</td><td>othch-m-otsi</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td>othmots da athi</td><td>= 80 + 10.</td></tr> +<tr><td>100.</td><td colspan="2">asi.</td></tr> +<tr><td>1000.</td><td>ath-asi</td><td>= 10 × 100.</td></tr> +</table> + +<table class="scale"> +<caption>Lazi</caption> + +<tr><td>10.</td><td colspan="2">wit.</td></tr> +<tr><td>20.</td><td colspan="2">öts.</td></tr> +<tr><td>30.</td><td>öts do wit</td><td>= 20 × 10.</td></tr> +<tr><td>40.</td><td>dzur en öts</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>dzur en öts do wit</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>dzum en öts</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>dzum en öts do wit</td><td>= 60 + 10.</td></tr> +<tr><td>80.</td><td>otch-an-öts</td><td>= 4 × 20.</td></tr> +<tr><td>100.</td><td colspan="2">os.</td></tr> +<tr><td>1000.</td><td colspan="2">silia (akin to Greek).</td></tr> +</table> + +<table class="scale"> +<caption>Chunsag.<a id="FNA-350" href="#FN-350"><sup>350</sup></a></caption> + +<tr><td>10.</td><td colspan="2">ants-go.</td></tr> +<tr><td>20.</td><td colspan="2">chogo.</td></tr> +<tr><td>30.</td><td>chogela antsgo</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>kichogo</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>kichelda antsgo</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>taw chago</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>taw chogelda antsgo</td><td>= 60 + 10.</td></tr> +<tr><td>80.</td><td>uch' chogo</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td colspan="2">uch' chogelda antsgo.</td></tr> +<tr><td>100.</td><td colspan="2">nusgo.</td></tr> +<tr><td>1000.</td><td colspan="2">asargo (akin to Persian).</td></tr> +</table> + +<table class="scale"> +<caption>Dido.<a id="FNA-351" href="#FN-351"><sup>351</sup></a></caption> + +<tr><td>10.</td><td colspan="2">zino.</td></tr> +<tr><td>20.</td><td colspan="2">ku.</td></tr> +<tr><td>30.</td><td colspan="2">kunozino.</td></tr> +<tr><td>40.</td><td>kaeno ku</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>kaeno kuno zino</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>sonno ku</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>sonno kuno zino</td><td>= 60 + 10.</td></tr> +<tr><td>80.</td><td>uino ku</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td>uino huno zino</td><td>= 80 + 10.</td></tr> +<tr><td>100.</td><td colspan="2">bischon.</td></tr> +<tr><td>400.</td><td>kaeno kuno zino</td><td>= 40 × 10.</td></tr> +</table> + + +<table class="scale"> +<caption><a class="pgnum" id="page-190" title="- 190 -"></a>Akari</caption> + +<tr><td>10.</td><td colspan="2">entzelgu.</td></tr> +<tr><td>20.</td><td colspan="2">kobbeggu.</td></tr> +<tr><td>30.</td><td colspan="2">lowergu.</td></tr> +<tr><td>40.</td><td>kokawu</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>kikaldanske</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td colspan="2">secikagu.</td></tr> +<tr><td>70.</td><td>kawalkaldansku</td><td>= 3 × 20 + 10.</td></tr> +<tr><td>80.</td><td>onkuku</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td>onkordansku</td><td>= 4 × 20 + 10.</td></tr> +<tr><td>100.</td><td colspan="2">nosku.</td></tr> +<tr><td>1000.</td><td colspan="2">askergu (from Persian).</td></tr> +</table> + +<table class="scale"> +<caption>Circassia</caption> + +<tr><td>10.</td><td colspan="2">psche.</td></tr> +<tr><td>20.</td><td colspan="2">to-tsch.</td></tr> +<tr><td>30.</td><td>totsch-era-pschirre</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>ptl'i-sch</td><td>= 4 × 10.</td></tr> +<tr><td>50.</td><td>ptl'isch-era-pschirre</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>chi-tsch</td><td>= 6 × 10.</td></tr> +<tr><td>70.</td><td>chitsch-era-pschirre</td><td>= 60 + 10.</td></tr> +<tr><td>80.</td><td>toshitl</td><td>= 20 × 4?</td></tr> +<tr><td>90.</td><td>toshitl-era-pschirre</td><td>= 80 + 10.</td></tr> +<tr><td>100.</td><td colspan="2">scheh.</td></tr> +<tr><td>1000.</td><td>min (Tartar) or schi-psche</td><td>= 100 × 10.</td></tr> +</table> + +<p>The last of these scales is an unusual combination of +decimal and vigesimal. In the even tens it is quite +regularly decimal, unless 80 is of the structure suggested +above. On the other hand, the odd tens are +formed in the ordinary vigesimal manner. The reason +for this anomaly is not obvious. I know of no other +number system that presents the same peculiarity, and +cannot give any hypothesis which will satisfactorily +account for its presence here. In nearly all the examples +given the decimal becomes the leading element +in the formation of all units above 100, just as was +the case in the Celtic scales already noticed.</p> + +<p>Among the northern tribes of Siberia the numeral +scales appear to be ruder and less simple than those +just examined, and the counting to be more consistently +<a class="pgnum" id="page-191" title="- 191 -"></a>vigesimal than in any scale we have thus far met +with. The two following examples are exceedingly interesting, +as being among the best illustrations of counting +by twenties that are to be found anywhere in the +Old World.</p> + +<table class="scale"> +<caption>Tschukschi.<a id="FNA-352" href="#FN-352"><sup>352</sup></a></caption> + +<tr><td>10.</td><td>migitken</td><td>= both hands.</td></tr> +<tr><td>20.</td><td>chlik-kin</td><td>= a whole man.</td></tr> +<tr><td>30.</td><td>chlikkin mingitkin parol</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>nirach chlikkin</td><td>= 2 × 20.</td></tr> +<tr><td>100.</td><td>milin chlikkin</td><td>= 5 × 20.</td></tr> +<tr><td>200.</td><td>mingit chlikkin</td><td>= 10 × 20, <i>i.e.</i> 10 men.</td></tr> +<tr><td>1000.</td><td>miligen chlin-chlikkin</td><td>= 5 × 200, <i>i.e.</i> five (times) 10 men.</td></tr> +</table> + +<table class="scale"> +<caption>Aino.<a id="FNA-353" href="#FN-353"><sup>353</sup></a></caption> + +<tr><td>10.</td><td colspan="2">wambi.</td></tr> +<tr><td>20.</td><td colspan="2">choz.</td></tr> +<tr><td>30.</td><td>wambi i-doehoz</td><td>= 10 from 40.</td></tr> +<tr><td>40.</td><td>tochoz</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>wambi i-richoz</td><td>= 10 from 60.</td></tr> +<tr><td>60.</td><td>rechoz</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>wambi [i?] inichoz</td><td>= 10 from 80.</td></tr> +<tr><td>80.</td><td>inichoz</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td>wambi aschikinichoz</td><td>= 10 from 100.</td></tr> +<tr><td>100.</td><td>aschikinichoz</td><td>= 5 × 20.</td></tr> +<tr><td>110.</td><td>wambi juwanochoz</td><td>= 10 from 120.</td></tr> +<tr><td>120.</td><td>juwano choz</td><td>= 6 × 20.</td></tr> +<tr><td>130.</td><td>wambi aruwanochoz</td><td>= 10 from 140.</td></tr> +<tr><td>140.</td><td>aruwano choz</td><td>= 7 × 20.</td></tr> +<tr><td>150.</td><td>wambi tubischano choz</td><td>= 10 from 160.</td></tr> +<tr><td>160.</td><td>tubischano choz</td><td>= 8 × 20.</td></tr> +<tr><td>170.</td><td>wambi schnebischano choz</td><td>= 10 from 180.</td></tr> +<tr><td>180.</td><td>schnebischano choz</td><td>= 9 × 20.</td></tr> +<tr><td><a class="pgnum" id="page-192" title="- 192 -"></a>190.</td><td>wambi schnewano choz</td><td>= 10 from 200.</td></tr> +<tr><td>200.</td><td>schnewano choz</td><td>= 10 × 20.</td></tr> +<tr><td>300.</td><td>aschikinichoz i gaschima chnewano choz</td><td>= 5 × 20 + 10 × 20.</td></tr> +<tr><td>400.</td><td>toschnewano choz</td><td>= 2 × (10 × 20).</td></tr> +<tr><td>500.</td><td>aschikinichoz i gaschima toschnewano choz</td><td>= 100 + 400.</td></tr> +<tr><td>600.</td><td>reschiniwano choz</td><td>= 3 × 200.</td></tr> +<tr><td>700.</td><td>aschikinichoz i gaschima reschiniwano choz</td><td>= 100 + 600.</td></tr> +<tr><td>800.</td><td>inischiniwano choz</td><td>= 4 × 200.</td></tr> +<tr><td>900.</td><td>aschikinichoz i gaschima inischiniwano choz</td><td>= 100 + 800.</td></tr> +<tr><td>1000.</td><td>aschikini schinewano choz</td><td>= 5 × 200.</td></tr> +<tr><td>2000.</td><td>wanu schinewano choz</td><td>= 10 × (10 × 20).</td></tr> +</table> + + +<p>This scale is in one sense wholly vigesimal, and in +another way it is not to be regarded as pure, but as +mixed. Below 20 it is quinary, and, however far it +might be extended, this quinary element would remain, +making the scale quinary-vigesimal. But in another +sense, also, the Aino system is not pure. In any unmixed +vigesimal scale the word for 400 must be a +simple word, and that number must be taken as the +vigesimal unit corresponding to 100 in the decimal +scale. But the Ainos have no simple numeral word +for any number above 20, forming all higher numbers +by combinations through one or more of the processes +of addition, subtraction, and multiplication. The only +number above 20 which is used as a unit is 200, which +is expressed merely as 10 twenties. Any even number +of hundreds, or any number of thousands, is then +indicated as being so many times 10 twenties; and +the odd hundreds are so many times 10 twenties, plus +5 twenties more. This scale is an excellent example +<a class="pgnum" id="page-193" title="- 193 -"></a>of the cumbersome methods used by uncivilized races +in extending their number systems beyond the ordinary +needs of daily life.</p> + +<p>In Central Asia a single vigesimal scale comes to +light in the following fragment of the Leptscha scale, +of the Himalaya region:<a id="FNA-354" href="#FN-354"><sup>354</sup></a></p> + +<table class="scale"> +<tr><td>10.</td><td colspan="2">kati.</td></tr> +<tr><td>40.</td><td>kafali</td><td>= 4 × 10,</td></tr> +<tr><td></td><td class="alt">or kha nat</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>kafano</td><td>= 5 × 10,</td></tr> +<tr><td></td><td class="alt">or kha nat sa kati</td><td>= 2 × 20 + 10.</td></tr> +<tr><td>100.</td><td colspan="2">gjo, or kat.</td></tr> +</table> + +<p>Further to the south, among the Dravidian races, the +vigesimal element is also found. The following will +suffice to illustrate the number systems of these dialects, +which, as far as the material at hand shows, are +different from each other only in minor particulars:</p> + +<table class="scale"> +<caption>Mundari.<a id="FNA-355" href="#FN-355"><sup>355</sup></a></caption> + +<tr><td>10.</td><td colspan="2">gelea.</td></tr> +<tr><td>20.</td><td colspan="2">mi hisi.</td></tr> +<tr><td>30.</td><td>mi hisi gelea</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>bar hisi</td><td>= 2 × 20.</td></tr> +<tr><td>60.</td><td>api hisi</td><td>= 3 × 20.</td></tr> +<tr><td>80.</td><td>upun hisi</td><td>= 4 × 20.</td></tr> +<tr><td>100.</td><td>mone hisi</td><td>= 5 × 20.</td></tr> +</table> + +<p>In the Nicobar Islands of the Indian Ocean a well-developed +example of vigesimal numeration is found. +The inhabitants of these islands are so low in the scale +of civilization that a definite numeral system of any +kind is a source of some surprise. Their neighbours, +the Andaman Islanders, it will be remembered, have +but two numerals at their command; their intelligence +<a class="pgnum" id="page-194" title="- 194 -"></a>does not seem in any way inferior to that of the Nicobar +tribes, and one is at a loss to account for the +superior development of the number sense in the case +of the latter. The intercourse of the coast tribes with +traders might furnish an explanation of the difficulty +were it not for the fact that the numeration of the inland +tribes is quite as well developed as that of the +coast tribes; and as the former never come in contact +with traders and never engage in barter of any kind +except in the most limited way, the conclusion seems +inevitable that this is merely one of the phenomena of +mental development among savage races for which we +have at present no adequate explanation. The principal +numerals of the inland and of the coast tribes are:<a id="FNA-356" href="#FN-356"><sup>356</sup></a></p> + +<div class="lefthalf"> +<table class="scale"> +<caption>Inland Tribes</caption> + +<tr><td>10.</td><td colspan="2">teya.</td></tr> +<tr><td>20.</td><td colspan="2">heng-inai.</td></tr> +<tr><td>30.</td><td>heng-inai-tain</td><td>= 20 + 5 (couples).</td></tr> +<tr><td>40.</td><td>au-inai</td><td>= 2 × 20.</td></tr> +<tr><td>100.</td><td>tain-inai</td><td>= 5 × 20.</td></tr> +<tr><td>200.</td><td>teya-inai</td><td>= 10 × 20.</td></tr> +<tr><td>300.</td><td>teya-tain-inai</td><td>= (10 + 5) × 20.</td></tr> +<tr><td>400.</td><td colspan="2">heng-teo.</td></tr> +</table> +</div> +<div class="righthalf"> +<table class="scale"> +<caption>Coast Tribes</caption> + +<tr><td>10.</td><td colspan="2">sham.</td></tr> +<tr><td>20.</td><td colspan="2">heang-inai.</td></tr> +<tr><td>30.</td><td>heang-inai-tanai</td><td>= 20 + 5 (couples).</td></tr> +<tr><td>40.</td><td>an-inai</td><td>= 2 × 20.</td></tr> +<tr><td>100.</td><td>tanai-inai</td><td>= 5 × 20.</td></tr> +<tr><td>200.</td><td>sham-inai</td><td>= 10 × 20.</td></tr> +<tr><td>300.</td><td>heang-tanai-inai</td><td>= (10 + 5) 20.</td></tr> +<tr><td>400.</td><td colspan="2">heang-momchiama.</td></tr> +</table> +</div> + +<p>In no other part of the world is vigesimal counting +found so perfectly developed, and, among native races, +so generally preferred, as in North and South America. +<a class="pgnum" id="page-195" title="- 195 -"></a>In the eastern portions of North America and in +the extreme western portions of South America the +decimal or the quinary decimal scale is in general +use. But in the northern regions of North America, in +western Canada and northwestern United States, in +Mexico and Central America, and in the northern and +western parts of South America, the unit of counting +among the great majority of the native races was 20. +The ethnological affinities of these races are not yet +definitely ascertained; and it is no part of the scope of +this work to enter into any discussion of that involved +question. But either through contact or affinity, this +form of numeration spread in prehistoric times over +half or more than half of the western hemisphere. It +was the method employed by the rude Eskimos of the +north and their equally rude kinsmen of Paraguay and +eastern Brazil; by the forest Indians of Oregon and +British Columbia, and by their more southern kinsmen, +the wild tribes of the Rio Grande and of the Orinoco. +And, most striking and interesting of all, it was the +method upon which were based the numeral systems of +the highly civilized races of Mexico, Yucatan, and New +Granada. Some of the systems obtained from the languages +of these peoples are perfect, extended examples +of vigesimal counting, not to be duplicated in any +other quarter of the globe. The ordinary unit was, as +would be expected, “one man,” and in numerous languages +the words for 20 and man are identical. But +in other cases the original meaning of that numeral +<a class="pgnum" id="page-196" title="- 196 -"></a>word has been lost; and in others still it has a signification +quite remote from that given above. These +meanings will be noticed in connection with the scales +themselves, which are given, roughly speaking, in their +geographical order, beginning with the Eskimo of the +far north. The systems of some of the tribes are as +follows:</p> + +<table class="scale"> +<caption>Alaskan Eskimos.<a id="FNA-357" href="#FN-357"><sup>357</sup></a></caption> + +<tr><td>10.</td><td colspan="2">koleet.</td></tr> +<tr><td>20.</td><td colspan="2">enuenok.</td></tr> +<tr><td>30.</td><td>enuenok kolinik</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>malho kepe ak</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>malho-kepe ak-kolmik che pah ak to</td><td>= 2 × 20 + 10.</td></tr> +<tr><td>60.</td><td>pingi shu-kepe ak</td><td>= 3 × 20.</td></tr> +<tr><td>100.</td><td>tale ma-kepe ak</td><td>= 5 × 20.</td></tr> +<tr><td>400.</td><td>enue nok ke pe ak</td><td>= 20 × 20.</td></tr> +</table> + +<table class="scale"> +<caption>Tchiglit.<a id="FNA-358" href="#FN-358"><sup>358</sup></a></caption> + +<tr><td>10.</td><td colspan="2">krolit.</td></tr> +<tr><td>20.</td><td>kroleti, or innun</td><td>= man.</td></tr> +<tr><td>30.</td><td>innok krolinik-tchikpalik</td><td>= man + 2 hands.</td></tr> +<tr><td>40.</td><td>innum mallerok</td><td>= 2 men.</td></tr> +<tr><td>50.</td><td>adjigaynarmitoat</td><td>= as many times 10 as the fingers of the hand.</td></tr> +<tr><td>60.</td><td>innumipit</td><td>= 3 men.</td></tr> +<tr><td>70.</td><td>innunmalloeronik arveneloerit</td><td>= 7 men?</td></tr> +<tr><td>80.</td><td>innun pinatçunik arveneloerit</td><td>= 8 men?</td></tr> +<tr><td>90.</td><td>innun tcitamanik arveneloerit</td><td>= 9 men?</td></tr> +<tr><td>100.</td><td colspan="2">itchangnerkr.</td></tr> +<tr><td>1000.</td><td>itchangner-park</td><td>= great 100.</td></tr> +</table> + +<p>The meanings for 70, 80, 90, are not given by Father +Petitot, but are of such a form that the significations +<a class="pgnum" id="page-197" title="- 197 -"></a>seem to be what are given above. Only a full acquaintance +with the Tchiglit language would justify one in +giving definite meanings to these words, or in asserting +that an error had been made in the numerals. But it +is so remarkable and anomalous to find the decimal +and vigesimal scales mingled in this manner that one +involuntarily suspects either incompleteness of form, or +an actual mistake.</p> + +<table class="scale"> +<caption>Tlingit.<a id="FNA-359" href="#FN-359"><sup>359</sup></a></caption> + +<tr><td>10.</td><td>djinkat</td><td>= both hands?</td></tr> +<tr><td>20.</td><td>tle ka</td><td>= 1 man.</td></tr> +<tr><td>30.</td><td>natsk djinkat</td><td>= 3 × 10.</td></tr> +<tr><td>40.</td><td>dak'on djinkat</td><td>= 4 × 10.</td></tr> +<tr><td>50.</td><td>kedjin djinkat</td><td>= 5 × 10.</td></tr> +<tr><td>60.</td><td>tle durcu djinkat</td><td>= 6 × 10.</td></tr> +<tr><td>70.</td><td>daqa durcu djinkat</td><td>= 7 × 10.</td></tr> +<tr><td>80.</td><td>natska durcu djinkat</td><td>= 8 × 10.</td></tr> +<tr><td>90.</td><td>gocuk durcu djinkat</td><td>= 9 × 10.</td></tr> +<tr><td>100.</td><td>kedjin ka</td><td>= 5 men, or 5 × 20.</td></tr> +<tr><td>200.</td><td>djinkat ka</td><td>= 10 × 20.</td></tr> +<tr><td>300.</td><td>natsk djinkat ka</td><td>= 30 men.</td></tr> +<tr><td>400.</td><td>dak'on djinkat ka</td><td>= 40 men.</td></tr> +</table> + +<p>This scale contains a strange commingling of decimal +and vigesimal counting. The words for 20, 100, and +200 are clear evidence of vigesimal, while 30 to 90, and +the remaining hundreds, are equally unmistakable proof +of decimal, numeration. The word <i>ka</i>, man, seems to +mean either 10 or 20; a most unusual occurrence. +The fact that a number system is partly decimal and +partly vigesimal is found to be of such frequent occurrence +<a class="pgnum" id="page-198" title="- 198 -"></a>that this point in the Tlingit scale need excite +no special wonder. But it is remarkable that the same +word should enter into numeral composition under such +different meanings.</p> + +<table class="scale"> +<caption>Nootka.<a id="FNA-360" href="#FN-360"><sup>360</sup></a></caption> + +<tr><td>10.</td><td colspan="2">haiu.</td></tr> +<tr><td>20.</td><td colspan="2">tsakeits.</td></tr> +<tr><td>30.</td><td>tsakeits ic haiu</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>atlek</td><td>= 2 × 20.</td></tr> +<tr><td>60.</td><td>katstsek</td><td>= 3 × 20.</td></tr> +<tr><td>80.</td><td>moyek</td><td>= 4 × 20.</td></tr> +<tr><td>100.</td><td>sutc'ek</td><td>= 5 × 20.</td></tr> +<tr><td>120.</td><td>nop'ok</td><td>= 6 × 20.</td></tr> +<tr><td>140.</td><td>atlpok</td><td>= 7 × 20.</td></tr> +<tr><td>160.</td><td>atlakutlek</td><td>= 8 × 20.</td></tr> +<tr><td>180.</td><td>ts'owakutlek</td><td>= 9 × 20.</td></tr> +<tr><td>200.</td><td>haiuk</td><td>= 10 × 20.</td></tr> +</table> + +<p>This scale is quinary-vigesimal, with no apparent +decimal element in its composition. But the derivation +of some of the terms used is detected with difficulty. +In the following scale the vigesimal structure is still +more obscure.</p> + +<table class="scale"> +<caption>Tsimshian.<a id="FNA-361" href="#FN-361"><sup>361</sup></a></caption> + +<tr><td>10.</td><td colspan="2">gy'ap.</td></tr> +<tr><td>20.</td><td>kyedeel</td><td>= 1 man.</td></tr> +<tr><td>30.</td><td colspan="2">gulewulgy'ap.</td></tr> +<tr><td>40.</td><td colspan="2">t'epqadalgyitk, or tqalpqwulgyap.</td></tr> +<tr><td>50.</td><td colspan="2">kctoncwulgyap.</td></tr> +<tr><td>100.</td><td colspan="2">kcenecal.</td></tr> +<tr><td>200.</td><td colspan="2">k'pal.</td></tr> +<tr><td>300.</td><td>k'pal te kcenecal</td><td>= 200 + 100.</td></tr> +<tr><td>400.</td><td colspan="2">kyedal.</td></tr> +<tr><td>500.</td><td>kyedal te kcenecal</td><td>= 400 + 100.</td></tr> +<tr><td>600.</td><td colspan="2">gulalegyitk.</td></tr> +<tr><td>700.</td><td>gulalegyitk te kcenecal</td><td>= 600 + 100.</td></tr> +<tr><td>800.</td><td colspan="2">tqalpqtalegyitk.</td></tr> +<tr><td>900.</td><td>tqalpqtalegyitk te kcenecal</td><td>= 800 + 100.</td></tr> +<tr><td>1000.</td><td colspan="2">k'pal.</td></tr> +</table> + +<p>To the unobservant eye this scale would certainly +appear to contain no more than a trace of the vigesimal +<a class="pgnum" id="page-199" title="- 199 -"></a>in its structure. But Dr. Boas, who is one of +the most careful and accurate of investigators, says in +his comment on this system: “It will be seen at once +that this system is quinary-vigesimal.… In 20 we +find the word <i>gyat</i>, man. The hundreds are identical +with the numerals used in counting men (see <a href="#page-87">p. 87</a>), +and then the quinary-vigesimal system is most evident.”</p> + +<table class="scale"> +<caption>Rio Norte Indians.<a id="FNA-362" href="#FN-362"><sup>362</sup></a></caption> + +<tr><td>20.</td><td colspan="2">taiguaco.</td></tr> +<tr><td>30.</td><td>taiguaco co juyopamauj ajte</td><td>= 20 + 2 × 5.</td></tr> +<tr><td>40.</td><td>taiguaco ajte</td><td>= 20 × 2.</td></tr> +<tr><td>50.</td><td>taiguaco ajte co juyopamauj ajte</td><td>= 20 × 2 + 5 × 2.</td></tr> +</table> + +<table class="scale"> +<caption>Caribs of Essiquibo, Guiana</caption> + +<tr><td>10.</td><td colspan="2">oween-abatoro.</td></tr> +<tr><td>20.</td><td>owee-carena</td><td>= 1 person.</td></tr> +<tr><td>40.</td><td>oko-carena</td><td>= 2 persons.</td></tr> +<tr><td>60.</td><td>oroowa-carena</td><td>= 3 persons.</td></tr> +</table> + +<table class="scale"> +<caption>Otomi</caption> + +<tr><td>10.</td><td colspan="2">ra-tta.</td></tr> +<tr><td>20.</td><td colspan="2">na-te.</td></tr> +<tr><td>30.</td><td>na-te-m'a-ratta</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>yo-te</td><td>= 2 × 30.</td></tr> +<tr><td>50.</td><td>yote-m'a-ratta</td><td>= 2 × 20 + 10.</td></tr> +<tr><td>60.</td><td>hiu-te</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>hiute-m'a-ratta</td><td>= 3 × 20 + 10.</td></tr> +<tr><td>80.</td><td>gooho-rate</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td>gooho-rate-m'a ratta</td><td>= 4 × 20 + 10.</td></tr> +<tr><td>100.</td><td>cytta-te</td><td>= 5 × 20,</td></tr> +<tr><td></td><td class="alt">or nanthebe</td><td>= 1 × 100.</td></tr> +</table> + +<table class="scale"> +<caption>Maya, Yucatan.<a id="FNA-363" href="#FN-363"><sup>363</sup></a></caption> + +<tr><td>1.</td><td colspan="2">hun.</td></tr> +<tr><td>10.</td><td>lahun</td><td>= it is finished.</td></tr> +<tr><td>20.</td><td>hunkal</td><td>= a measure, or more correctly, a fastening together.</td></tr> +<tr><td>30.</td><td>lahucakal</td><td>= 40 − 10?</td></tr> +<tr><td>40.</td><td>cakal</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>lahuyoxkal</td><td>= 60 − 10.</td></tr> +<tr><td>60.</td><td>oxkal</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>lahucankal</td><td>= 80 − 10.</td></tr> +<tr><td>80.</td><td>cankal</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td>lahuyokal</td><td>= 100 − 10.</td></tr> +<tr><td>100.</td><td>hokal</td><td>= 5 × 20.</td></tr> +<tr><td>110.</td><td>lahu uackal</td><td>= 120 − 10.</td></tr> +<tr><td>120.</td><td>uackal</td><td>= 6 × 20.</td></tr> +<tr><td>130.</td><td>lahu uuckal</td><td>= 140 − 10.</td></tr> +<tr><td><a class="pgnum" id="page-200" title="- 200 -"></a>140.</td><td>uuckal</td><td>= 7 × 20.</td></tr> +<tr><td>200.</td><td>lahuncal</td><td>= 10 × 20.</td></tr> +<tr><td>300.</td><td>holhukal</td><td>= 15 × 20.</td></tr> +<tr><td>400.</td><td>hunbak</td><td>= 1 tying around.</td></tr> +<tr><td>500.</td><td colspan="2">hotubak.</td></tr> +<tr><td>600.</td><td colspan="2">lahutubak</td></tr> +<tr><td>800.</td><td>calbak</td><td>= 2 × 400.</td></tr> +<tr><td>900.</td><td colspan="2">hotu yoxbak.</td></tr> +<tr><td>1000.</td><td colspan="2">lahuyoxbak.</td></tr> +<tr><td>1200.</td><td>oxbak</td><td>= 3 × 400.</td></tr> +<tr><td>2000.</td><td colspan="2">capic (modern).</td></tr> +<tr><td>8000.</td><td>hunpic</td><td>= 1 sack.</td></tr> +<tr><td>16,000.</td><td colspan="2">ca pic (ancient).</td></tr> +<tr><td>160,000.</td><td>calab</td><td>= a filling full</td></tr> +<tr><td>3,200,000.</td><td colspan="2">kinchil.</td></tr> +<tr><td>64,000,000.</td><td colspan="2">hunalau.</td></tr> +</table> + +<p>In the Maya scale we have one of the best and most +extended examples of vigesimal numeration ever developed +by any race. To show in a more striking and forcible +manner the perfect regularity of the system, the +following tabulation is made of the various Maya units, +which will correspond to the “10 units make one ten, +10 tens make one hundred, 10 hundreds make one thousand,” +etc., which old-fashioned arithmetic compelled us +to learn in childhood. The scale is just as regular by +twenties in Maya as by tens in English. It is<a id="FNA-364" href="#FN-364"><sup>364</sup></a></p> + +<table class="units"> +<tr><td>20 hun</td><td>= 1 kal</td><td>= 20.</td></tr> +<tr><td>20 kal</td><td>= 1 bak</td><td>= 400.</td></tr> +<tr><td>20 bak</td><td>= 1 pic</td><td>= 8000.</td></tr> +<tr><td>20 pic</td><td>= 1 calab</td><td>= 160,000.</td></tr> +<tr><td style="vertical-align:middle">20 calab</td><td><table class="layout"><tr><td rowspan="2">= 1<span class="twoline">{</span></td><td>kinchil</td><td rowspan="2"><span class="twoline">}</span></td></tr><tr><td>tzotzceh</td></tr></table></td><td style="vertical-align:middle">= 3,200,000.</td></tr> +<tr><td>20 kinchil</td><td>= 1 alau</td><td>= 64,000,000.</td></tr> +</table> + +<p>The original meaning of <i>pic</i>, given in the scale as +“a sack,” was rather “a short petticoat, somtimes used +as a sack.” The word <i>tzotzceh</i> signified “deerskin.” +No reason can be given for the choice of this word as +a numeral, though the appropriateness of the others is +<a class="pgnum" id="page-201" title="- 201 -"></a>sufficiently manifest. No evidence of digital numeration +appears in the first 10 units, but, judging from +the almost universal practice of the Indian tribes of +both North and South America, such may readily have +been the origin of Maya counting. Whatever its origin, +it certainly expanded and grew into a system whose +perfection challenges our admiration. It was worthy of +the splendid civilization of this unfortunate race, and, +through its simplicity and regularity, bears ample testimony +to the intellectual capacity which originated it.</p> + +<p>The only example of vigesimal reckoning which is comparable +with that of the Mayas is the system employed +by their northern neighbours, the Nahuatl, or, as they are +more commonly designated, the Aztecs of Mexico. This +system is quite as pure and quite as simple as the Maya, +but differs from it in some important particulars. In +its first 20 numerals it is quinary (see <a href="#page-141">p. 141</a>), and as +a system must be regarded as quinary-vigesimal. The +Maya scale is decimal through its first 20 numerals, +and, if it is to be regarded as a mixed scale, must +be characterized as decimal-vigesimal. But in both +these instances the vigesimal element preponderates so +strongly that these, in common with their kindred number +systems of Mexico, Yucatan, and Central America, +are always thought of and alluded to as vigesimal +scales. On account of its importance, the Nahuatl system<a id="FNA-365" href="#FN-365"><sup>365</sup></a> +is given in fuller detail than most of the other +systems I have made use of.</p> + +<table class="scale"> +<tr><td><a class="pgnum" id="page-202" title="- 202 -"></a>10.</td><td>matlactli</td><td>= 2 hands.</td></tr> +<tr><td>20.</td><td>cempoalli</td><td>= 1 counting.</td></tr> +<tr><td>21.</td><td>cempoalli once</td><td>= 20-1.</td></tr> +<tr><td>22.</td><td>cempoalli omome</td><td>= 20-2.</td></tr> +<tr><td>30.</td><td>cempoalli ommatlactli</td><td>= 20-10.</td></tr> +<tr><td>31.</td><td>cempoalli ommatlactli once</td><td>= 20-10-1.</td></tr> +<tr><td>40.</td><td>ompoalli</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>ompoalli ommatlactli</td><td>= 40-10.</td></tr> +<tr><td>60.</td><td>eipoalli, or epoalli,</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>epoalli ommatlactli</td><td>= 60-10.</td></tr> +<tr><td>80.</td><td>nauhpoalli</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td>nauhpoalli ommatlactli</td><td>= <del title="arithmetically wrong">90</del><ins title="replaced 90 with 80 for the sake of arithmetical correctness and in agreement with previous numerals">80</ins>-10.</td></tr> +<tr><td>100.</td><td>macuilpoalli</td><td>= 5 × 20.</td></tr> +<tr><td>120.</td><td>chiquacempoalli</td><td>= 6 × 20.</td></tr> +<tr><td>140.</td><td>chicompoalli</td><td>= 7 × 20.</td></tr> +<tr><td>160.</td><td>chicuepoalli</td><td>= 8 × 20.</td></tr> +<tr><td>180.</td><td>chiconauhpoalli</td><td>= 9 × 20.</td></tr> +<tr><td>200.</td><td>matlacpoalli</td><td>= 10 × 20.</td></tr> +<tr><td>220.</td><td>matlactli oncempoalli</td><td>= 11 × 20.</td></tr> +<tr><td>240.</td><td>matlactli omompoalli</td><td>= 12 × 20.</td></tr> +<tr><td>260.</td><td>matlactli omeipoalli</td><td>= 13 × 20.</td></tr> +<tr><td>280.</td><td>matlactli onnauhpoalli</td><td>= 14 × 20.</td></tr> +<tr><td>300.</td><td>caxtolpoalli</td><td>= 15 × 20.</td></tr> +<tr><td>320.</td><td colspan="2">caxtolli oncempoalli.</td></tr> +<tr><td>399.</td><td>caxtolli onnauhpoalli ipan caxtolli onnaui</td><td>= 19 × 20 + 19.</td></tr> +<tr><td>400.</td><td>centzontli</td><td>= 1 bunch of grass, or 1 tuft of hair.</td></tr> +<tr><td>800.</td><td>ometzontli</td><td>= 2 × 400.</td></tr> +<tr><td>1200.</td><td>eitzontli</td><td>= 3 × 400.</td></tr> +<tr><td>7600.</td><td>caxtolli onnauhtzontli</td><td>= 19 × 400.</td></tr> +<tr><td>8000.</td><td colspan="2">cenxiquipilli, or cexiquipilli.</td></tr> +<tr><td>160,000.</td><td>cempoalxiquipilli</td><td>= 20 × 8000.</td></tr> +<tr><td>3,200,000.</td><td>centzonxiquipilli</td><td>= 400 × 8000.</td></tr> +<tr><td>64,000,000.</td><td>cempoaltzonxiquipilli</td><td>= 20 × 400 × 8000.</td></tr> +</table> + +<p>Up to 160,000 the Nahuatl system is as simple and +regular in its construction as the English. But at this +<a class="pgnum" id="page-203" title="- 203 -"></a>point it fails in the formation of a new unit, or rather +in the expression of its new unit by a simple word; +and in the expression of all higher numbers it is forced +to resort in some measure to compound terms, just as +the English might have done had it not been able to +borrow from the Italian. The higher numeral terms, +under such conditions, rapidly become complex and cumbersome, +as the following analysis of the number 1,279,999,999 +shows.<a id="FNA-366" href="#FN-366"><sup>366</sup></a> The analysis will be readily understood +when it is remembered that <i>ipan</i> signifies plus. <i>Caxtolli +onnauhpoaltzonxiquipilli ipan caxtolli onnauhtzonxiquipilli +ipan caxtolli onnauhpoalxiquipilli ipan caxtolli onnauhxiquipilli +ipan caxtolli onnauhtzontli ipan caxtolli onnauhpoalli +ipan caxtolli onnaui;</i> <i>i.e.</i> 1,216,000,000 + 60,800,000 + + 3,040,000 + 152,000 + 7600 + 380 + 19. To show the +compounding which takes place in the higher numerals, +the analysis may be made more literally, thus: + <ins title="term added for the sake of arithmetical correctness, cf. numerals given">(15 + 4) × 20 × 400 × 8000 </ins>+ (15 + 4) × 400 × 800<ins title="appended a zero for the sake of arithmetical correctness, cf. numerals given">0</ins> + (15 + 4) × 20 × 8000 + (15 + + 4) × 8000 + (15 + 4) × 400 + (15 + 4) × 20 + 15 + + 4. Of course this resolution suffers from the fact +that it is given in digits arranged in accordance with +decimal notation, while the Nahuatl numerals express +values by a base twice as great. This gives the effect +of a complexity and awkwardness greater than really +existed in the actual use of the scale. Except for the +presence of the quinary element the number just given +<a class="pgnum" id="page-204" title="- 204 -"></a>is really expressed with just as great simplicity as it +could be in English words if our words “million” and +“billion” were replaced by “thousand thousand” and +“thousand thousand thousand.” If Mexico had remained +undisturbed by Europeans, and science and commerce +had been left to their natural growth and development, +uncompounded words would undoubtedly have been +found for the higher units, 160,000, 3,200,000, etc., +and the system thus rendered as simple as it is possible +for a quinary-vigesimal system to be.</p> + +<p>Other number scales of this region are given as +follows:</p> + +<table class="scale"> +<caption>Huasteca.<a id="FNA-367" href="#FN-367"><sup>367</sup></a></caption> + +<tr><td>10.</td><td colspan="2">laluh.</td></tr> +<tr><td>20.</td><td>hum-inic</td><td>= 1 man.</td></tr> +<tr><td>30.</td><td>hum-inic-lahu</td><td>= 1 man 10.</td></tr> +<tr><td>40.</td><td>tzab-inic</td><td>= 2 men.</td></tr> +<tr><td>50.</td><td>tzab-inic-lahu</td><td>= 2 men 10.</td></tr> +<tr><td>60.</td><td>ox-inic</td><td>= 3 men.</td></tr> +<tr><td>70.</td><td>ox-inic-lahu</td><td>= 3 men 10.</td></tr> +<tr><td>80.</td><td>tze-tnic</td><td>= 4 men.</td></tr> +<tr><td>90.</td><td>tze-ynic-kal-laluh</td><td>= 4 men and 10.</td></tr> +<tr><td>100.</td><td>bo-inic</td><td>= 5 men.</td></tr> +<tr><td>200.</td><td>tzab-bo-inic</td><td>= 2 × 5 men.</td></tr> +<tr><td>300.</td><td>ox-bo-inic</td><td>= 3 × 5 men.</td></tr> +<tr><td>400.</td><td>tsa-bo-inic</td><td>= 4 × 5 men.</td></tr> +<tr><td>600.</td><td>acac-bo-inic</td><td>= 6 × 5 men.</td></tr> +<tr><td>800.</td><td>huaxic-bo-inic</td><td>= 8 × 5 men.</td></tr> +<tr><td>1000.</td><td colspan="2">xi.</td></tr> +<tr><td>8000.</td><td>huaxic-xi</td><td>= 8-1000.</td></tr> +</table> + +<p>The essentially vigesimal character of this system +changes in the formation of some of the higher numerals, +and a suspicion of the decimal enters. One hundred is +<i>boinic</i>, 5 men; but 200, instead of being simply <i>lahuh-inic</i>, +10 men, is <i>tsa-bo-inic</i>, 2 × 100, or more strictly, 2 times +5 men. Similarly, 300 is 3 × 100, 400 is 4 × 100, etc. +<a class="pgnum" id="page-205" title="- 205 -"></a>The word for 1000 is simple instead of compound, and +the thousands appear to be formed wholly on the decimal +base. A comparison of this scale with that of the +Nahuatl shows how much inferior it is to the latter, +both in simplicity and consistency.</p> + +<table class="scale"> +<caption>Totonaco.<a id="FNA-368" href="#FN-368"><sup>368</sup></a></caption> + +<tr><td>10.</td><td colspan="2">cauh.</td></tr> +<tr><td>20.</td><td colspan="2">puxam.</td></tr> +<tr><td>30.</td><td>puxamacauh</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>tipuxam</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>tipuxamacauh</td><td>= 40 + 10.</td></tr> +<tr><td>60.</td><td>totonpuxam</td><td>= 3 × 20.</td></tr> +<tr><td>100.</td><td>quitziz puxum</td><td>= 5 × 20.</td></tr> +<tr><td>200.</td><td>copuxam</td><td>= 10 × 20.</td></tr> +<tr><td>400.</td><td colspan="2">tontaman.</td></tr> +<tr><td>1000.</td><td>titamanacopuxam</td><td>= 2 × 400 + 200.</td></tr> +</table> + +<p>The essential character of the vigesimal element is +shown by the last two numerals. <i>Tontamen</i>, the square +of 20, is a simple word, and 1000 is, as it should be, +2 times 400, plus 200. It is most unfortunate that the +numeral for 8000, the cube of 20, is not given.</p> + +<table class="scale"> +<caption>Cora.<a id="FNA-369" href="#FN-369"><sup>369</sup></a></caption> + +<tr><td>10.</td><td colspan="2">tamoamata.</td></tr> +<tr><td>20.</td><td colspan="2">cei-tevi.</td></tr> +<tr><td>30.</td><td>ceitevi apoan tamoamata</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>huapoa-tevi</td><td>= 2 × 20.</td></tr> +<tr><td>60.</td><td>huaeica-tevi</td><td>= 3 × 20.</td></tr> +<tr><td>100.</td><td>anxu-tevi</td><td>= 5 × 20.</td></tr> +<tr><td>400.</td><td>ceitevi-tevi</td><td>= 20 × 20.</td></tr> +</table> + +<p>Closely allied with the Maya numerals and method +of counting are those of the Quiches of Guatemala. The +resemblance is so obvious that no detail in the Quiche +scale calls for special mention.</p> + +<table class="scale"> +<caption><a class="pgnum" id="page-206" title="- 206 -"></a>Quiche.<a id="FNA-370" href="#FN-370"><sup>370</sup></a></caption> + +<tr><td>10.</td><td colspan="2">lahuh.</td></tr> +<tr><td>20.</td><td>hu-uinac</td><td>= 1 man.</td></tr> +<tr><td>30.</td><td>hu-uinac-lahuh</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>ca-uinac</td><td>= 2 men.</td></tr> +<tr><td>50.</td><td>lahu-r-ox-kal</td><td>= −10 + 3 × 20.</td></tr> +<tr><td>60.</td><td>ox-kal</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>lahu-u-humuch</td><td>= −10 + 80.</td></tr> +<tr><td>80.</td><td colspan="2">humuch.</td></tr> +<tr><td>90.</td><td>lahu-r-ho-kal</td><td>= −10 + 100.</td></tr> +<tr><td>100.</td><td colspan="2">hokal.</td></tr> +<tr><td>1000.</td><td colspan="2">o-tuc-rox-o-kal.</td></tr> +</table> + +<p>Among South American vigesimal systems, the best +known is that of the Chibchas or Muyscas of the Bogota +region, which was obtained at an early date by the missionaries +who laboured among them. This system is +much less extensive than that of some of the more +northern races; but it is as extensive as almost any +other South American system with the exception of the +Peruvian, which was, however, a pure decimal system. +As has already been stated, the native races of South +America were, as a rule, exceedingly deficient in regard to +the number sense. Their scales are rude, and show great +poverty, both in formation of numeral words and in the +actual extent to which counting was carried. If extended +as far as 20, these scales are likely to become vigesimal, +but many stop far short of that limit, and no inconsiderable +number of them fail to reach even 5. In this +respect we are reminded of the Australian scales, which +were so rudimentary as really to preclude any proper +use of the word “system” in connection with them. +Counting among the South American tribes was often +<a class="pgnum" id="page-207" title="- 207 -"></a>equally limited, and even less regular. Following are +the significant numerals of the scale in question:</p> + +<table class="scale"> +<caption>Chibcha, or Muysca.<a id="FNA-371" href="#FN-371"><sup>371</sup></a></caption> + +<tr><td>10.</td><td colspan="2">hubchibica.</td></tr> +<tr><td>20.</td><td>quihica ubchihica</td><td>= thus says the foot, 10 = 10-10,</td></tr> +<tr><td></td><td class="alt">or gueta</td><td>= house.</td></tr> +<tr><td>30.</td><td>guetas asaqui ubchihica</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>gue-bosa</td><td>= 20 × 2.</td></tr> +<tr><td>60.</td><td>gue-mica</td><td>= 20 × 3.</td></tr> +<tr><td>80.</td><td>gue-muyhica</td><td>= 20 × 4.</td></tr> +<tr><td>100.</td><td>gue-hisca</td><td>= 20 × 5.</td></tr> +</table> + +<table class="scale"> +<caption>Nagranda.<a id="FNA-372" href="#FN-372"><sup>372</sup></a></caption> + +<tr><td>10.</td><td colspan="2">guha.</td></tr> +<tr><td>20.</td><td colspan="2">dino.</td></tr> +<tr><td>30.</td><td>'badiñoguhanu</td><td>= 20 + 10.</td></tr> +<tr><td>40.</td><td>apudiño</td><td>= 2 × 20.</td></tr> +<tr><td>50.</td><td>apudiñoguhanu</td><td>= 2 × 20 + 10.</td></tr> +<tr><td>60.</td><td>asudiño</td><td>= 3 × 20.</td></tr> +<tr><td>70.</td><td>asudiñoguhanu</td><td>= 3 × 20 + 10.</td></tr> +<tr><td>80.</td><td>acudiño</td><td>= 4 × 20.</td></tr> +<tr><td>90.</td><td>acudiñoguhanu</td><td>= 4 × 20 + 10.</td></tr> +<tr><td>100.</td><td>huisudiño</td><td>= 5 × 20,</td></tr> +<tr><td></td><td class="alt">or guhamba</td><td>= great 10.</td></tr> +<tr><td>200.</td><td>guahadiño</td><td>= 10 × 20.</td></tr> +<tr><td>400.</td><td>diñoamba</td><td>= great 20.</td></tr> +<tr><td>1000.</td><td>guhaisudiño</td><td>= 10 × 5 × 20.</td></tr> +<tr><td>2000.</td><td>hisudiñoamba</td><td>= 5 great 20's.</td></tr> +<tr><td>4000.</td><td>guhadiñoamba</td><td>= 10 great 20's.</td></tr> +</table> + +<p>In considering the influence on the manners and customs +of any people which could properly be ascribed +to the use among them of any other base than 10, it +must not be forgotten that no races, save those using +that base, have ever attained any great degree of civilization, +with the exception of the ancient Aztecs and +their immediate neighbours, north and south. For reasons +already pointed out, no highly civilized race has +ever used an exclusively quinary system; and all that +can be said of the influence of this mode of counting +<a class="pgnum" id="page-208" title="- 208 -"></a>is that it gives rise to the habit of collecting objects +in groups of five, rather than of ten, when any attempt +is being made to ascertain their sum. In the case of +the subsidiary base 12, for which the Teutonic races +have always shown such a fondness, the dozen and +gross of commerce, the divisions of English money, and +of our common weights and measures are probably an +outgrowth of this preference; and the Babylonian base, +60, has fastened upon the world forever a sexagesimal +method of dividing time, and of measuring the circumference +of the circle.</p> + +<p>The advanced civilization attained by the races of +Mexico and Central America render it possible to see +some of the effects of vigesimal counting, just as a +single thought will show how our entire lives are influenced +by our habit of counting by tens. Among the +Aztecs the universal unit was 20. A load of cloaks, of +dresses, or other articles of convenient size, was 20. +Time was divided into periods of 20 days each. The +armies were numbered by divisions of 8000;<a id="FNA-373" href="#FN-373"><sup>373</sup></a> and in +countless other ways the vigesimal element of numbers +entered into their lives, just as the decimal enters into +ours; and it is to be supposed that they found it as +useful and as convenient for all measuring purposes as +we find our own system; as the tradesman of to-day +finds the duodecimal system of commerce; or as the +Babylonians of old found that singularly curious system, +the sexagesimal. Habituation, the laws which the +<a class="pgnum" id="page-209" title="- 209 -"></a>habits and customs of every-day life impose upon us, +are so powerful, that our instinctive readiness to make +use of any concept depends, not on the intrinsic +perfection or imperfection which pertains to it, but +on the familiarity with which previous use has invested +it. Hence, while one race may use a decimal, +another a quinary-vigesimal, and another a sexagesimal +scale, and while one system may actually be inherently +superior to another, no user of one method of reckoning +need ever think of any other method as possessing +practical inconveniences, of which those employing it +are ever conscious. And, to cite a single instance +which illustrates the unconscious daily use of two +modes of reckoning in one scale, we have only to think +of the singular vigesimal fragment which remains to +this day imbedded in the numeral scale of the French. +In counting from 70 to 100, or in using any number +which lies between those limits, no Frenchman is conscious +of employing a method of numeration less simple +or less convenient in any particular, than when he +is at work with the strictly decimal portions of his +scale. He passes from the one style of counting to the +other, and from the second back to the first again, +entirely unconscious of any break or change; entirely +unconscious, in fact, that he is using any particular +system, except that which the daily habit of years has +made a part himself.</p> + +<p>Deep regret must be felt by every student of philology, +that the primitive meanings of simple numerals +<a class="pgnum" id="page-210" title="- 210 -"></a>have been so generally lost. But, just as the pebble +on the beach has been worn and rounded by the beating +of the waves and by other pebbles, until no trace +of its original form is left, and until we can say of it +now only that it is quartz, or that it is diorite, so too +the numerals of many languages have suffered from the +attrition of the ages, until all semblance of their origin +has been lost, and we can say of them only that +they are numerals. Beyond a certain point we can +carry the study neither of number nor of number +words. At that point both the mathematician and the +philologist must pause, and leave everything beyond to +the speculations of those who delight in nothing else +so much as in pure theory.</p> + + + + +<h3>The End.</h3> + + + + + +<h2><a class="pgnum" id="page-211" title="- 211 -"></a>Index Of Authors.</h2> + + +<ul class="index"> +<li class="newletter">Adam, L., <span><a href="#page-44">44</a>, <a href="#page-159">159</a>, <a href="#page-166">166</a>, <a href="#page-175">175</a>.</span></li> +<li>Armstrong, R. A., <span><a href="#page-180">180</a>.</span></li> +<li>Aymonier, A., <span><a href="#page-156">156</a>.</span></li> + + +<li class="newletter">Bachofen, J. J., <span><a href="#page-131">131</a>.</span></li> +<li>Balbi, A., <span><a href="#page-151">151</a>.</span></li> +<li>Bancroft, H. H., <span><a href="#page-29">29</a>, <a href="#page-47">47</a>, <a href="#page-89">89</a>, <a href="#page-93">93</a>, <a href="#page-113">113</a>, <a href="#page-199">199</a>.</span></li> +<li>Barlow, H., <span><a href="#page-108">108</a>.</span></li> +<li>Beauregard, O., <span><a href="#page-45">45</a>, <a href="#page-83">83</a>, <a href="#page-152">152</a>.</span></li> +<li>Bellamy, E. W., <span><a href="#page-9">9</a>.</span></li> +<li>Boas, F., <span><a href="#page-30">30</a>, <a href="#page-45">45</a>, <a href="#page-46">46</a>, <a href="#page-65">65</a>, <a href="#page-87">87</a>, <a href="#page-88">88</a>, <a href="#page-136">136</a>, <a href="#page-163">163</a>, <a href="#page-164">164</a>, <a href="#page-171">171</a>, <a href="#page-197">197</a>, <a href="#page-198">198</a>.</span></li> +<li>Bonwick, J., <span><a href="#page-24">24</a>, <a href="#page-27">27</a>, <a href="#page-107">107</a>, <a href="#page-108">108</a>.</span></li> +<li>Brinton, D. G., <span><a href="#page-2">2</a>, <a href="#page-22">22</a>, <a href="#page-46">46</a>, <a href="#page-52">52</a>, <a href="#page-57">57</a>, <a href="#page-61">61</a>, <a href="#page-111">111</a>, <a href="#page-112">112</a>, <a href="#page-140">140</a>, <a href="#page-199">199</a>, <a href="#page-200">200</a>.</span></li> +<li>Burton, R. F., <span><a href="#page-37">37</a>, <a href="#page-71">71</a>.</span></li> + + +<li class="newletter">Chamberlain, A. F., <span><a href="#page-45">45</a>, <a href="#page-65">65</a>, <a href="#page-93">93</a>.</span></li> +<li>Chase, P. E., <span><a href="#page-99">99</a>.</span></li> +<li>Clarke, H., <span><a href="#page-113">113</a>.</span></li> +<li>Codrington, R. H., <span><a href="#page-16">16</a>, <a href="#page-95">95</a>, <a href="#page-96">96</a>, <a href="#page-136">136</a>, <a href="#page-138">138</a>, <a href="#page-145">145</a>, <a href="#page-153">153</a>, <a href="#page-154">154</a>.</span></li> +<li>Crawfurd, J., <span><a href="#page-89">89</a>, <a href="#page-93">93</a>, <a href="#page-130">130</a>.</span></li> +<li>Curr, E. M., <span><a href="#page-24">24–27</a>, <a href="#page-104">104</a>, <a href="#page-107">107–110</a>, <a href="#page-112">112</a>.</span></li> +<li>Cushing, F. H., <span><a href="#page-13">13</a>, <a href="#page-48">48</a>.</span></li> + + +<li class="newletter">De Flacourt, <span><a href="#page-8">8</a>, <a href="#page-9">9</a>.</span></li> +<li>De Quincey, T., <span><a href="#page-35">35</a>.</span></li> +<li>Deschamps, M., <span><a href="#page-28">28</a>.</span></li> +<li>Dobrizhoffer, M., <span><a href="#page-71">71</a>.</span></li> +<li>Dorsey, J. O., <span><a href="#page-59">59</a>.</span></li> +<li>Du Chaillu, P. B., <span><a href="#page-66">66</a>, <a href="#page-67">67</a>, <a href="#page-150">150</a>, <a href="#page-151">151</a>.</span></li> +<li>Du Graty, A. M., <span><a href="#page-138">138</a>.</span></li> + + +<li class="newletter">Ellis, A. A., <span><a href="#page-64">64</a>, <a href="#page-91">91</a>.</span></li> +<li>Ellis, R., <span><a href="#page-37">37</a>, <a href="#page-142">142</a>.</span></li> +<li>Ellis, W., <span><a href="#page-83">83</a>, <a href="#page-119">119</a>.</span></li> +<li>Erskine, J. E., <span><a href="#page-153">153</a>, <a href="#page-154">154</a>.</span></li> + + +<li class="newletter">Flegel, R., <span><a href="#page-133">133</a>.</span></li> + + +<li class="newletter">Gallatin, A., <span><a href="#page-136">136</a>, <a href="#page-159">159</a>, <a href="#page-166">166</a>, <a href="#page-171">171</a>, <a href="#page-199">199</a>, <a href="#page-204">204</a>, <a href="#page-206">206</a>, <a href="#page-208">208</a>.</span></li> +<li>Galton, F., <span><a href="#page-4">4</a>.</span></li> +<li>Gatschet, A. S., <span><a href="#page-58">58</a>, <a href="#page-59">59</a>, <a href="#page-68">68</a>.</span></li> +<li>Gilij, F. S., <span><a href="#page-54">54</a>.</span></li> +<li>Gill, W. W., <span><a href="#page-18">18</a>, <a href="#page-118">118</a>.</span></li> +<li>Goedel, M., <span><a href="#page-83">83</a>, <a href="#page-147">147</a>.</span></li> +<li>Grimm, J. L. C., <span><a href="#page-48">48</a>.</span></li> +<li>Gröber, G., <span><a href="#page-182">182</a>.</span></li> +<li>Guillome, J., <span><a href="#page-181">181</a>.</span></li> + + +<li class="newletter">Haddon, A. C., <span><a href="#page-18">18</a>, <a href="#page-105">105</a>.</span></li> +<li>Hale, H., <span><a href="#page-61">61</a>, <a href="#page-65">65</a>, <a href="#page-93">93</a>, <a href="#page-114">114–116</a>, <a href="#page-122">122</a>, <a href="#page-130">130</a>, <a href="#page-156">156</a>, <a href="#page-163">163</a>, <a href="#page-164">164</a>, <a href="#page-171">171</a>.</span></li> +<li>Hankel, H., <span><a href="#page-137">137</a>.</span></li> +<li>Haumonté, J. D., <span><a href="#page-44">44</a>.</span></li> +<li>Hervas, L., <span><a href="#page-170">170</a>.</span></li> +<li>Humboldt, A. von, <span><a href="#page-32">32</a>, <a href="#page-207">207</a>.</span></li> +<li>Hyades, M., <span><a href="#page-22">22</a>.</span></li> + + +<li class="newletter">Kelly, J. W., <span><a href="#page-157">157</a>, <a href="#page-196">196</a>.</span></li> +<li>Kelly, J., <span><a href="#page-180">180</a>.</span></li> +<li>Kleinschmidt, S., <span><a href="#page-52">52</a>, <a href="#page-80">80</a>.</span></li> + + +<li class="newletter">Lang, J. D., <span><a href="#page-108">108</a>.</span></li> +<li>Lappenberg, J. M., <span><a href="#page-127">127</a>.</span></li> +<li>Latham, R. G., <span><a href="#page-24">24</a>, <a href="#page-67">67</a>, <a href="#page-107">107</a>.</span></li> +<li>Leibnitz, G. W. von, <span><a href="#page-102">102</a>, <a href="#page-103">103</a>.</span></li> +<li>Lloyd, H. E., <span><a href="#page-7">7</a>.</span></li> +<li>Long, C. C., <span><a href="#page-148">148</a>, <a href="#page-186">186</a>.</span></li> +<li>Long, S. H., <span><a href="#page-121">121</a>.</span></li> +<li>Lubbock, Sir J., <span><a href="#page-3">3</a>, <a href="#page-5">5</a>.</span></li> +<li>Lull, E. P., <span><a href="#page-79">79</a>.</span></li> + + +<li class="newletter">Macdonald, J., <span><a href="#page-15">15</a>.</span></li> +<li>Mackenzie, A., <span><a href="#page-26">26</a>.</span></li> +<li>Man, E. H., <span><a href="#page-28">28</a>, <a href="#page-194">194</a>.</span></li> +<li>Mann, A., <span><a href="#page-47">47</a>.</span></li> +<li>Marcoy, P. (Saint Cricq), <span><a href="#page-23">23</a>, <a href="#page-168">168</a>.</span></li> +<li>Mariner, A., <span><a href="#page-85">85</a>.</span></li> +<li><a class="pgnum" id="page-212" title="- 212 -"></a>Martius, C. F. von, <span><a href="#page-23">23</a>, <a href="#page-79">79</a>, <a href="#page-111">111</a>, <a href="#page-122">122</a>, <a href="#page-138">138</a>, <a href="#page-142">142</a>, <a href="#page-174">174</a>.</span></li> +<li>Mason, <span><a href="#page-112">112</a>.</span></li> +<li>Mill, J. S., <span><a href="#page-3">3</a>.</span></li> +<li>Moncelon, M., <span><a href="#page-142">142</a>.</span></li> +<li>Morice, A., <span><a href="#page-15">15</a>, <a href="#page-86">86</a>.</span></li> +<li>Müller, Fr., <span><a href="#page-10">10</a>, <a href="#page-27">27</a>, <a href="#page-28">28</a>, <a href="#page-45">45</a>, <a href="#page-48">48</a>, <a href="#page-55">55</a>, <a href="#page-56">56</a>, <a href="#page-60">60</a>, <a href="#page-63">63</a>, <a href="#page-66">66</a>, <a href="#page-69">69</a>, <a href="#page-78">78</a>, <a href="#page-80">80</a>, <a href="#page-90">90</a>, <a href="#page-108">108</a>, <a href="#page-111">111</a>, <a href="#page-121">121</a>, <a href="#page-122">122</a>, <a href="#page-130">130</a>, <a href="#page-136">136</a>, <a href="#page-139">139</a>, <a href="#page-146">146–151</a>, <a href="#page-156">156–158</a>, <a href="#page-165">165–167</a>, <a href="#page-185">185–187</a>, <a href="#page-191">191</a>, <a href="#page-193">193</a>.</span></li> +<li>Murdoch, J., <span><a href="#page-30">30</a>, <a href="#page-49">49</a>,<a href="#page-137">137</a>.</span></li> + + +<li class="newletter">Nystron, J. W., <span><a href="#page-132">132</a>.</span></li> + + +<li class="newletter">O'Donovan, J., <span><a href="#page-180">180</a>.</span></li> +<li>Oldfield, A., <span><a href="#page-29">29</a>, <a href="#page-77">77</a>.</span></li> +<li>Olmos, A. de, <span><a href="#page-141">141</a>.</span></li> + + +<li class="newletter">Parisot, J., <span><a href="#page-44">44</a>.</span></li> +<li>Park, M., <span><a href="#page-145">145–147</a>.</span></li> +<li>Parry, W. E., <span><a href="#page-32">32</a>.</span></li> +<li>Peacock, G., <span><a href="#page-8">8</a>, <a href="#page-56">56</a>, <a href="#page-84">84</a>, <a href="#page-111">111</a>, <a href="#page-118">118</a>, <a href="#page-119">119</a>, <a href="#page-154">154</a>, <a href="#page-186">186</a>.</span></li> +<li>Petitot, E., <span><a href="#page-53">53</a>, <a href="#page-157">157</a>, <a href="#page-196">196</a>.</span></li> +<li>Pott, A. F., <span><a href="#page-50">50</a>, <a href="#page-68">68</a>, <a href="#page-92">92</a>, <a href="#page-120">120</a>, <a href="#page-145">145</a>, <a href="#page-148">148</a>, <a href="#page-149">149</a>, <a href="#page-152">152</a>, <a href="#page-157">157</a>, <a href="#page-166">166</a>, <a href="#page-182">182</a>, <a href="#page-184">184</a>, <a href="#page-189">189</a>, <a href="#page-191">191</a>, <a href="#page-205">205</a>.</span></li> +<li>Pruner-Bey, <span><a href="#page-10">10</a>, <a href="#page-104">104</a>.</span></li> +<li>Pughe, W. O., <span><a href="#page-141">141</a>.</span></li> + + +<li class="newletter">Ralph, J., <span><a href="#page-125">125</a>.</span></li> +<li>Ray, S. H., <span><a href="#page-45">45</a>, <a href="#page-78">78</a>, <a href="#page-80">80</a>.</span></li> +<li>Ridley, W., <span><a href="#page-57">57</a>.</span></li> +<li>Roth, H. L., <span><a href="#page-79">79</a>.</span></li> + + +<li class="newletter">Salt, H., <span><a href="#page-187">187</a>.</span></li> +<li>Sayce, A. H., <span><a href="#page-75">75</a>.</span></li> +<li>Schoolcraft, H. R., <span><a href="#page-66">66</a>, <a href="#page-81">81</a>, <a href="#page-83">83</a>, <a href="#page-84">84</a>, <a href="#page-159">159</a>, <a href="#page-160">160</a>.</span></li> +<li>Schröder, P., <span><a href="#page-90">90</a>.</span></li> +<li>Schweinfurth, G., <span><a href="#page-143">143</a>, <a href="#page-146">146</a>, <a href="#page-149">149</a>, <a href="#page-186">186</a>, <a href="#page-187">187</a>.</span></li> +<li>Simeon, R., <span><a href="#page-201">201</a>.</span></li> +<li>Spix, J. B. von, <span><a href="#page-7">7</a>.</span></li> +<li>Spurrell, W., <span><a href="#page-180">180</a>.</span></li> +<li>Squier, G. E., <span><a href="#page-80">80</a>, <a href="#page-207">207</a>.</span></li> +<li>Stanley, H. M., <span><a href="#page-38">38</a>, <a href="#page-42">42</a>, <a href="#page-64">64</a>, <a href="#page-69">69</a>, <a href="#page-78">78</a>, <a href="#page-150">150</a>, <a href="#page-187">187</a>.</span></li> + + +<li class="newletter">Taplin, G., <span><a href="#page-106">106</a>.</span></li> +<li>Thiel, B. A., <span><a href="#page-172">172</a>.</span></li> +<li>Toy, C. H., <span><a href="#page-70">70</a>.</span></li> +<li>Turner, G., <span><a href="#page-152">152</a>, <a href="#page-154">154</a>.</span></li> +<li>Tylor, E. B., <span><a href="#page-2">2</a>, <a href="#page-3">3</a>, <a href="#page-15">15</a>, <a href="#page-18">18</a>, <a href="#page-22">22</a>, <a href="#page-63">63</a>, <a href="#page-65">65</a>, <a href="#page-78">78</a>, <a href="#page-79">79</a>, <a href="#page-81">81</a>, <a href="#page-84">84</a>, <a href="#page-97">97</a>, <a href="#page-124">124</a>.</span></li> + + +<li class="newletter">Van Eys, J. W., <span><a href="#page-182">182</a>.</span></li> +<li>Vignoli, T., <span><a href="#page-95">95</a>.</span></li> + + +<li class="newletter">Wallace, A. R., <span><a href="#page-174">174</a>.</span></li> +<li>Wells, E. R., jr., <span><a href="#page-157">157</a>, <a href="#page-196">196</a>.</span></li> +<li>Whewell, W., <span><a href="#page-3">3</a>.</span></li> +<li>Wickersham, J., <span><a href="#page-96">96</a>.</span></li> +<li>Wiener, C., <span><a href="#page-22">22</a>.</span></li> +<li>Williams, W. L., <span><a href="#page-123">123</a>.</span></li> +</ul> + + + + + +<h2><a class="pgnum" id="page-213" title="- 213 -"></a>Index Of Subjects.</h2> + + +<ul class="index"> +<li class="newletter">Abacus, <span><a href="#page-19">19</a>.</span></li> +<li>Abeokuta, <span><a href="#page-33">33</a>.</span></li> +<li>Abipone, <span><a href="#page-71">71</a>, <a href="#page-72">72</a>.</span></li> +<li>Abkhasia, <span><a href="#page-188">188</a>.</span></li> +<li>Aboker, <span><a href="#page-148">148</a>.</span></li> +<li>Actuary, Life ins., <span><a href="#page-19">19</a>.</span></li> +<li>Adaize, <span><a href="#page-162">162</a>.</span></li> +<li>Addition, <span><a href="#page-19">19</a>, <a href="#page-43">43</a>, <a href="#page-46">46</a>, <a href="#page-92">92</a>.</span></li> +<li>Adelaide, <span><a href="#page-108">108</a>.</span></li> +<li>Admiralty Islands, <span><a href="#page-45">45</a>.</span></li> +<li>Affadeh, <span><a href="#page-184">184</a>.</span></li> +<li>Africa (African), <span><a href="#page-9">9</a>, <a href="#page-16">16</a>, <a href="#page-28">28</a>, <a href="#page-29">29</a>, <a href="#page-32">32</a>, <a href="#page-33">33</a>, <a href="#page-38">38</a>, <a href="#page-42">42</a>, <a href="#page-47">47</a>, <a href="#page-64">64</a>, <a href="#page-66">66</a>, <a href="#page-69">69</a>, <a href="#page-78">78</a>, <a href="#page-80">80</a>, <a href="#page-91">91</a>, <a href="#page-105">105</a>, <a href="#page-120">120</a>, <a href="#page-145">145</a>, <a href="#page-170">170</a>, <a href="#page-176">176</a>, <a href="#page-184">184</a>, <a href="#page-187">187</a>.</span></li> +<li>Aino (Ainu), <span><a href="#page-45">45</a>, <a href="#page-191">191</a>.</span></li> +<li>Akra, <span><a href="#page-120">120</a>.</span></li> +<li>Akari, <span><a href="#page-190">190</a>.</span></li> +<li>Alaska, <span><a href="#page-157">157</a>, <a href="#page-196">196</a>.</span></li> +<li>Albania, <span><a href="#page-184">184</a>.</span></li> +<li>Albert River, <span><a href="#page-26">26</a>.</span></li> +<li>Aleut, <span><a href="#page-157">157</a>.</span></li> +<li>Algonkin (Algonquin), <span><a href="#page-45">45</a>, <a href="#page-92">92</a>, <a href="#page-161">161</a>.</span></li> +<li>Amazon, <span><a href="#page-23">23</a>.</span></li> +<li>Ambrym, <span><a href="#page-136">136</a>.</span></li> +<li>American, <span><a href="#page-10">10</a>, <a href="#page-16">16</a>, <a href="#page-19">19</a>, <a href="#page-98">98</a>, <a href="#page-105">105</a>.</span></li> +<li>Andaman, <span><a href="#page-8">8</a>, <a href="#page-15">15</a>, <a href="#page-28">28</a>, <a href="#page-31">31</a>, <a href="#page-76">76</a>, <a href="#page-174">174</a>, <a href="#page-193">193</a>.</span></li> +<li>Aneitum, <span><a href="#page-154">154</a>.</span></li> +<li>Animal, <span><a href="#page-3">3</a>, <a href="#page-6">6</a>.</span></li> +<li>Anthropological, <span><a href="#page-21">21</a>.</span></li> +<li>Apho, <span><a href="#page-133">133</a>.</span></li> +<li>Api, <span><a href="#page-80">80</a>, <a href="#page-136">136</a>, <a href="#page-155">155</a>.</span></li> +<li>Apinage, <span><a href="#page-111">111</a>.</span></li> +<li>Arab, <span><a href="#page-170">170</a>.</span></li> +<li>Arawak, <span><a href="#page-52">52–54</a>, <a href="#page-135">135</a>.</span></li> +<li>Arctic, <span><a href="#page-29">29</a>.</span></li> +<li>Arikara, <span><a href="#page-46">46</a>.</span></li> +<li>Arithmetic, <span><a href="#page-1">1</a>, <a href="#page-5">5</a>, <a href="#page-30">30</a>, <a href="#page-33">33</a>, <a href="#page-73">73</a>, <a href="#page-93">93</a>.</span></li> +<li>Aryan, 76, <span><a href="#page-128">128–130</a>.</span></li> +<li>Ashantee, <span><a href="#page-145">145</a>.</span></li> +<li>Asia (Asiatic), <span><a href="#page-28">28</a>, <a href="#page-113">113</a>, <a href="#page-131">131</a>, <a href="#page-187">187</a>.</span></li> +<li>Assiniboine, <span><a href="#page-66">66</a>, <a href="#page-92">92</a>.</span></li> +<li>Athapaskan,<span><a href="#page-92">92</a>.</span></li> +<li>Atlantic, <span><a href="#page-126">126</a>.</span></li> +<li>Aurora, <span><a href="#page-155">155</a>.</span></li> +<li>Australia (Australian), <span><a href="#page-2">2</a>, <a href="#page-6">6</a>, <a href="#page-19">19</a>, <a href="#page-22">22</a>, <a href="#page-24">24–30</a>, <a href="#page-57">57</a>, <a href="#page-58">58</a>, <a href="#page-71">71</a>, <a href="#page-75">75</a>, <a href="#page-76">76</a>, <a href="#page-84">84</a>, <a href="#page-103">103</a>, <a href="#page-105">105</a>, <a href="#page-106">106</a>, <a href="#page-110">110</a>, <a href="#page-112">112</a>, <a href="#page-118">118</a>, <a href="#page-173">173</a>, <a href="#page-206">206</a>.</span></li> +<li>Avari, <span><a href="#page-188">188</a>.</span></li> +<li>Aymara, <span><a href="#page-166">166</a>.</span></li> +<li>Aztec, <span><a href="#page-63">63</a>, <a href="#page-78">78</a>, <a href="#page-83">83</a>, <a href="#page-89">89</a>, <a href="#page-93">93</a>, <a href="#page-201">201</a>, <a href="#page-207">207</a>, <a href="#page-208">208</a>.</span></li> + + +<li class="newletter">Babusessé, <span><a href="#page-38">38</a>.</span></li> +<li>Babylonian, <span><a href="#page-208">208</a>.</span></li> +<li>Bagrimma, <span><a href="#page-148">148</a>.</span></li> +<li>Bahnars, <span><a href="#page-15">15</a>.</span></li> +<li>Bakairi, <span><a href="#page-111">111</a>.</span></li> +<li>Balad, <span><a href="#page-67">67</a>.</span></li> +<li>Balenque, <span><a href="#page-150">150</a>.</span></li> +<li>Bambarese, <span><a href="#page-95">95</a>.</span></li> +<li>Banks Islands, <span><a href="#page-16">16</a>, <a href="#page-96">96</a>, <a href="#page-153">153</a>.</span></li> +<li>Barea, <span><a href="#page-151">151</a>.</span></li> +<li>Bargaining, <span><a href="#page-18">18</a>, <a href="#page-19">19</a>, <a href="#page-32">32</a>.</span></li> +<li>Bari, <span><a href="#page-136">136</a>.</span></li> +<li>Barre, <span><a href="#page-174">174</a>.</span></li> +<li>Basa, <span><a href="#page-146">146</a>.</span></li> +<li>Basque, <span><a href="#page-40">40</a>, <a href="#page-182">182</a>.</span></li> +<li>Bellacoola, see Bilqula.</li> +<li>Belyando River, <span><a href="#page-109">109</a>.</span></li> +<li>Bengal, Bay of, <span><a href="#page-28">28</a>.</span></li> +<li>Benuë, <span><a href="#page-133">133</a>.</span></li> +<li>Betoya, <span><a href="#page-57">57</a>, <a href="#page-112">112</a>, <a href="#page-135">135</a>, <a href="#page-140">140</a>.</span></li> +<li>Bilqula, <span><a href="#page-46">46</a>, <a href="#page-164">164</a>.</span></li> +<li>Binary, <span><a href="#page-100">chap. v</a>.</span></li> +<li>Binin, <span><a href="#page-149">149</a>.</span></li> +<li>Bird-nesting, <span><a href="#page-5">5</a>.</span></li> +<li>Bisaye, <span><a href="#page-90">90</a>.</span></li> +<li>Bogota, <span><a href="#page-206">206</a>.</span></li> +<li>Bolan, <span><a href="#page-120">120</a>.</span></li> +<li>Bolivia, <span><a href="#page-2">2</a>, <a href="#page-21">21</a>.</span></li> +<li>Bongo, <span><a href="#page-143">143</a>, <a href="#page-186">186</a>.</span></li> +<li><a class="pgnum" id="page-214" title="- 214 -"></a>Bonzé, <span><a href="#page-151">151</a>.</span></li> +<li>Bororo, <span><a href="#page-23">23</a>.</span></li> +<li>Botocudo, <span><a href="#page-22">22</a>, <a href="#page-31">31</a>, <a href="#page-48">48</a>, <a href="#page-71">71</a>.</span></li> +<li>Bourke, <span><a href="#page-108">108</a>.</span></li> +<li>Boyne River, <span><a href="#page-24">24</a>.</span></li> +<li>Brazil, <span><a href="#page-2">2</a>, <a href="#page-7">7</a>, <a href="#page-30">30</a>, <a href="#page-174">174</a>, <a href="#page-195">195</a>.</span></li> +<li>Bretagne (Breton), <span><a href="#page-120">120</a>, <a href="#page-129">129</a>, <a href="#page-181">181</a>, <a href="#page-182">182</a>.</span></li> +<li>British Columbia, <span><a href="#page-45">45</a>, <a href="#page-46">46</a>, <a href="#page-65">65</a>, <a href="#page-86">86</a>, <a href="#page-88">88</a>, <a href="#page-89">89</a>, <a href="#page-112">112</a>, <a href="#page-113">113</a>, <a href="#page-195">195</a>.</span></li> +<li>Bullom, <span><a href="#page-147">147</a>.</span></li> +<li>Bunch, <span><a href="#page-64">64</a>.</span></li> +<li>Burnett River, <span><a href="#page-112">112</a>.</span></li> +<li>Bushman, <span><a href="#page-28">28</a>, <a href="#page-31">31</a>.</span></li> +<li>Butong, <span><a href="#page-93">93</a>.</span></li> + + +<li class="newletter">Caddoe, <span><a href="#page-162">162</a>.</span></li> +<li>Cahuillo, <span><a href="#page-165">165</a>.</span></li> +<li>Calculating machine, <span><a href="#page-19">19</a>.</span></li> +<li>Campa, <span><a href="#page-22">22</a>.</span></li> +<li>Canada, <span><a href="#page-29">29</a>, <a href="#page-53">53</a>, <a href="#page-54">54</a>, <a href="#page-86">86</a>, <a href="#page-195">195</a>.</span></li> +<li>Canaque, <span><a href="#page-142">142</a>, <a href="#page-144">144</a>.</span></li> +<li>Caraja, <span><a href="#page-23">23</a>.</span></li> +<li>Carib, <span><a href="#page-166">166</a>, <a href="#page-167">167</a>, <a href="#page-199">199</a>.</span></li> +<li>Carnarvon, <span><a href="#page-35">35</a>, <a href="#page-36">36</a>.</span></li> +<li>Carrier, <span><a href="#page-86">86</a>.</span></li> +<li>Carthaginian, <span><a href="#page-179">179</a>.</span></li> +<li>Caucasus, <span><a href="#page-188">188</a>.</span></li> +<li>Cayriri (see Kiriri), <span><a href="#page-79">79</a>.</span></li> +<li>Cayubaba (Cayubabi), <span><a href="#page-84">84</a>, <a href="#page-167">167</a>.</span></li> +<li>Celtic, <span><a href="#page-40">40</a>, <a href="#page-169">169</a>, <a href="#page-179">179</a>, <a href="#page-181">181</a>, <a href="#page-190">190</a>.</span></li> +<li>Cely, Mom, <span><a href="#page-9">9</a>.</span></li> +<li>Central America, <span><a href="#page-29">29</a>, <a href="#page-69">69</a>, <a href="#page-79">79</a>, <a href="#page-121">121</a>, <a href="#page-131">131</a>, <a href="#page-195">195</a>, <a href="#page-201">201</a>, <a href="#page-208">208</a>.</span></li> +<li>Ceylon, <span><a href="#page-28">28</a>.</span></li> +<li>Chaco, <span><a href="#page-22">22</a>.</span></li> +<li>Champion Bay, <span><a href="#page-109">109</a>.</span></li> +<li>Charles XII., <span><a href="#page-132">132</a>.</span></li> +<li>Cheyenne, <span><a href="#page-62">62</a>.</span></li> +<li>Chibcha, <span><a href="#page-206">206</a>.</span></li> +<li>China (Chinese), <span><a href="#page-40">40</a>, <a href="#page-131">131</a>.</span></li> +<li>Chippeway, <span><a href="#page-62">62</a>, <a href="#page-159">159</a>, <a href="#page-162">162</a>.</span></li> +<li>Chiquito, <span><a href="#page-2">2</a>, <a href="#page-6">6</a>, <a href="#page-21">21</a>, <a href="#page-40">40</a>, <a href="#page-71">71</a>, <a href="#page-76">76</a>.</span></li> +<li>Choctaw, <span><a href="#page-65">65</a>, <a href="#page-85">85</a>, <a href="#page-162">162</a>.</span></li> +<li>Chunsag, <span><a href="#page-189">189</a>.</span></li> +<li>Circassia, <span><a href="#page-190">190</a>.</span></li> +<li>Cobeu, <span><a href="#page-174">174</a>.</span></li> +<li>Cochin China, <span><a href="#page-15">15</a>.</span></li> +<li>Columbian, <span><a href="#page-113">113</a>.</span></li> +<li>Comanche, <span><a href="#page-29">29</a>, <a href="#page-83">83</a>.</span></li> +<li>Conibo, <span><a href="#page-23">23</a>.</span></li> +<li>Cooper's Creek, <span><a href="#page-108">108</a>.</span></li> +<li>Cora, <span><a href="#page-166">166</a>.</span></li> +<li>Cotoxo, <span><a href="#page-111">111</a>.</span></li> +<li>Cowrie, <span><a href="#page-64">64</a>, <a href="#page-70">70</a>, <a href="#page-71">71</a>.</span></li> +<li>Cree, <span><a href="#page-91">91</a>.</span></li> +<li>Crocker Island, <span><a href="#page-107">107</a>.</span></li> +<li>Crow, <span><a href="#page-3">3</a>, <a href="#page-4">4</a>, <a href="#page-92">92</a>.</span></li> +<li>Crusoe, Robinson, <span><a href="#page-7">7</a>.</span></li> +<li>Curetu, <span><a href="#page-111">111</a>.</span></li> + + +<li class="newletter">Dahomey, <span><a href="#page-71">71</a>.</span></li> +<li>Dakota, <span><a href="#page-81">81</a>, <a href="#page-91">91</a>, <a href="#page-92">92</a>.</span></li> +<li>Danish, <span><a href="#page-30">30</a>, <a href="#page-46">46</a>, <a href="#page-129">129</a>, <a href="#page-183">183</a>.</span></li> +<li>Darnley Islands, <span><a href="#page-24">24</a>.</span></li> +<li>Delaware, <span><a href="#page-91">91</a>, <a href="#page-160">160</a>.</span></li> +<li>Demara, <span><a href="#page-4">4</a>, <a href="#page-6">6</a>.</span></li> +<li>Déné, <span><a href="#page-86">86</a>.</span></li> +<li>Dido, <span><a href="#page-189">189</a>.</span></li> +<li>Dinka, <span><a href="#page-136">136</a>, <a href="#page-147">147</a>.</span></li> +<li>Dippil, <span><a href="#page-107">107</a>.</span></li> +<li>Division, <span><a href="#page-19">19</a>.</span></li> +<li>Dravidian, <span><a href="#page-104">104</a>, <a href="#page-193">193</a>.</span></li> +<li>Dual number, <span><a href="#page-75">75</a>.</span></li> +<li>Duluth, <span><a href="#page-34">34</a>.</span></li> +<li>Duodecimal, <span><a href="#page-100">chap. v</a>.</span></li> +<li>Dutch, <span><a href="#page-129">129</a>.</span></li> + + +<li class="newletter">Eaw, <span><a href="#page-24">24</a>.</span></li> +<li>Ebon, <span><a href="#page-152">152</a>.</span></li> +<li>Efik, <span><a href="#page-148">148</a>, <a href="#page-185">185</a>.</span></li> +<li>Encabellada, <span><a href="#page-22">22</a>.</span></li> +<li>Encounter Bay, <span><a href="#page-108">108</a>.</span></li> +<li>Ende, <span><a href="#page-68">68</a>, <a href="#page-152">152</a>.</span></li> +<li>English, <span><a href="#page-28">28</a>, <a href="#page-38">38–44</a>, <a href="#page-60">60</a>, <a href="#page-81">81</a>, <a href="#page-85">85</a>, <a href="#page-89">89</a>, <a href="#page-118">118</a>, <a href="#page-123">123</a>, <a href="#page-124">124</a>, <a href="#page-129">129</a>, <a href="#page-183">183</a>, <a href="#page-200">200</a>, <a href="#page-203">203</a>, <a href="#page-208">208</a>.</span></li> +<li>Eromanga, <span><a href="#page-96">96</a>, <a href="#page-136">136</a>, <a href="#page-154">154</a>.</span></li> +<li>Eskimo, <span><a href="#page-16">16</a>, <a href="#page-30">30</a>, <a href="#page-31">31</a>, <a href="#page-32">32</a>, <a href="#page-36">36</a>, <a href="#page-48">48</a>, <a href="#page-51">51</a>, <a href="#page-52">52</a>, <a href="#page-54">54</a>, <a href="#page-61">61</a>, <a href="#page-64">64</a>, <a href="#page-83">83</a>, <a href="#page-137">137</a>, <a href="#page-157">157</a>, <a href="#page-159">159</a>, <a href="#page-195">195</a>, <a href="#page-196">196</a>.</span></li> +<li>Essequibo, <span><a href="#page-166">166</a>.</span></li> +<li>Europe (European), <span><a href="#page-27">27</a>, <a href="#page-39">39</a>, <a href="#page-168">168</a>, <a href="#page-169">169</a>, <a href="#page-179">179</a>, <a href="#page-182">182</a>, <a href="#page-183">183</a>, <a href="#page-185">185</a>, <a href="#page-204">204</a>.</span></li> +<li>Eye, <span><a href="#page-14">14</a>, <a href="#page-97">97</a>.</span></li> +<li>Eyer's Sand Patch, <span><a href="#page-26">26</a>.</span></li> +<li>Ewe, <span><a href="#page-64">64</a>, <a href="#page-91">91</a>.</span></li> + + +<li class="newletter">Fall, <span><a href="#page-163">163</a>.</span></li> +<li>Fate, <span><a href="#page-138">138</a>, <a href="#page-155">155</a>.</span></li> +<li>Fatuhiva, <span><a href="#page-130">130</a>.</span></li> +<li>Feloop, <span><a href="#page-145">145</a>.</span></li> +<li>Fernando Po, <span><a href="#page-150">150</a>.</span></li> +<li>Fiji, <span><a href="#page-96">96</a>.</span></li> +<li>Finger pantomime, <span><a href="#page-10">10</a>, <a href="#page-23">23</a>, <a href="#page-29">29</a>, <a href="#page-67">67</a>, <a href="#page-82">82</a>.</span></li> +<li>Fingoe, <span><a href="#page-33">33</a>.</span></li> +<li>Fist, <span><a href="#page-16">16</a>, <a href="#page-59">59</a>, <a href="#page-72">72</a>.</span></li> +<li><a class="pgnum" id="page-215" title="- 215 -"></a>Flinder's River, <span><a href="#page-24">24</a>.</span></li> +<li>Flores, <span><a href="#page-68">68</a>, <a href="#page-152">152</a>.</span></li> +<li>Forefinger, <span><a href="#page-12">12</a>, <a href="#page-15">15</a>, <a href="#page-16">16</a>, <a href="#page-54">54</a>, <a href="#page-61">61</a>, <a href="#page-91">91</a>, <a href="#page-113">113</a>.</span></li> +<li>Foulah, <span><a href="#page-147">147</a>.</span></li> +<li>Fourth finger, <span><a href="#page-18">18</a>.</span></li> +<li>Frazer's Island, <span><a href="#page-108">108</a>.</span></li> +<li>French, <span><a href="#page-40">40</a>, <a href="#page-41">41</a>, <a href="#page-124">124</a>, <a href="#page-129">129</a>, <a href="#page-181">181</a>, <a href="#page-182">182</a>, <a href="#page-209">209</a>.</span></li> +<li>Fuegan, <span><a href="#page-22">22</a>.</span></li> + + +<li class="newletter">Gaelic, <span><a href="#page-180">180</a>.</span></li> +<li>Galibi, <span><a href="#page-138">138</a>.</span></li> +<li>Gaul, <span><a href="#page-169">169</a>, <a href="#page-182">182</a>.</span></li> +<li>Georgia, <span><a href="#page-189">189</a>.</span></li> +<li>German, <span><a href="#page-38">38–43</a>, <a href="#page-129">129</a>, <a href="#page-183">183</a>.</span></li> +<li>Gesture, <span><a href="#page-18">18</a>, <a href="#page-59">59</a>.</span></li> +<li>Gola, <span><a href="#page-151">151</a>.</span></li> +<li>Golo, <span><a href="#page-146">146</a>.</span></li> +<li>Gonn Station, <span><a href="#page-110">110</a>.</span></li> +<li>Goth, <span><a href="#page-169">169</a>.</span></li> +<li>Greek, <span><a href="#page-86">86</a>, <a href="#page-129">129</a>, <a href="#page-168">168</a>, <a href="#page-169">169</a>.</span></li> +<li>Green Island, <span><a href="#page-45">45</a>.</span></li> +<li>Greenland, <span><a href="#page-29">29</a>, <a href="#page-52">52</a>, <a href="#page-80">80</a>, <a href="#page-158">158</a>.</span></li> +<li>Guachi, <span><a href="#page-23">23</a>, <a href="#page-31">31</a>.</span></li> +<li>Guarani, <span><a href="#page-55">55</a>, <a href="#page-138">138</a>.</span></li> +<li>Guatemala, <span><a href="#page-205">205</a>.</span></li> +<li>Guato, <span><a href="#page-142">142</a>.</span></li> +<li>Guaycuru, <span><a href="#page-22">22</a>.</span></li> +<li>Gudang, <span><a href="#page-24">24</a>.</span></li> + + +<li class="newletter">Haida, <span><a href="#page-112">112</a>.</span></li> +<li>Hawaii, <span><a href="#page-113">113</a>, <a href="#page-114">114</a>, <a href="#page-116">116</a>, <a href="#page-117">117</a>.</span></li> +<li>Head, <span><a href="#page-71">71</a>.</span></li> +<li>Heap, <span><a href="#page-8">8</a>, <a href="#page-9">9</a>, <a href="#page-25">25</a>, <a href="#page-70">70</a>, <a href="#page-77">77</a>, <a href="#page-100">100</a>.</span></li> +<li>Hebrew, <span><a href="#page-86">86</a>, <a href="#page-89">89</a>, <a href="#page-95">95</a>.</span></li> +<li>Heiltsuk, <span><a href="#page-65">65</a>, <a href="#page-88">88</a>, <a href="#page-163">163</a>.</span></li> +<li>Herero, <span><a href="#page-150">150</a>.</span></li> +<li>Hervey Islands, <span><a href="#page-118">118</a>.</span></li> +<li>Hidatsa, <span><a href="#page-80">80</a>, <a href="#page-91">91</a>.</span></li> +<li>Hill End, <span><a href="#page-109">109</a>.</span></li> +<li>Himalaya, <span><a href="#page-193">193</a>.</span></li> +<li>Hottentot, <span><a href="#page-80">80</a>, <a href="#page-92">92</a>.</span></li> +<li>Huasteca, <span><a href="#page-204">204</a>.</span></li> +<li>Hudson's Bay, <span><a href="#page-48">48</a>, <a href="#page-61">61</a>.</span></li> +<li>Hun, <span><a href="#page-169">169</a>.</span></li> +<li>Hunt, Leigh, <span><a href="#page-33">33</a>.</span></li> + + +<li class="newletter">Ibo, <span><a href="#page-185">185</a>.</span></li> +<li>Icelandic, <span><a href="#page-129">129</a>, <a href="#page-183">183</a>.</span></li> +<li>Illinois, <span><a href="#page-91">91</a>.</span></li> +<li>Index finger, <span><a href="#page-11">11</a>, <a href="#page-14">14</a>.</span></li> +<li>India, <span><a href="#page-96">96</a>, <a href="#page-112">112</a>, <a href="#page-131">131</a>.</span></li> +<li>Indian, <span><a href="#page-8">8</a>, <a href="#page-10">10</a>, <a href="#page-13">13</a>, <a href="#page-16">16</a>, <a href="#page-17">17</a>, <a href="#page-19">19</a>, <a href="#page-32">32</a>, <a href="#page-36">36</a>, <a href="#page-54">54</a>, <a href="#page-55">55</a>, <a href="#page-59">59</a>, <a href="#page-62">62</a>, <a href="#page-65">65</a>, <a href="#page-66">66</a>, <a href="#page-79">79</a>, <a href="#page-80">80</a>, <a href="#page-82">82</a>, <a href="#page-83">83</a>, <a href="#page-89">89</a>, <a href="#page-90">90</a>, <a href="#page-98">98</a>, <a href="#page-105">105</a>, <a href="#page-112">112</a>, <a href="#page-171">171</a>, <a href="#page-201">201</a>.</span></li> +<li>Indian Ocean, <span><a href="#page-63">63</a>, <a href="#page-193">193</a>.</span></li> +<li>Indo-European, <span><a href="#page-76">76</a>.</span></li> +<li>Irish, <span><a href="#page-129">129</a>, <a href="#page-180">180</a>.</span></li> +<li>Italian, <span><a href="#page-39">39</a>, <a href="#page-80">80</a>, <a href="#page-124">124</a>, <a href="#page-129">129</a>, <a href="#page-203">203</a>.</span></li> + + +<li class="newletter">Jajowerong, <span><a href="#page-156">156</a>.</span></li> +<li>Jallonkas, <span><a href="#page-146">146</a>.</span></li> +<li>Jaloff, <span><a href="#page-146">146</a>.</span></li> +<li>Japanese, <span><a href="#page-40">40</a>, <a href="#page-86">86</a>, <a href="#page-89">89</a>, <a href="#page-93">93–95</a>.</span></li> +<li>Java, <span><a href="#page-93">93</a>, <a href="#page-120">120</a>.</span></li> +<li>Jiviro, <span><a href="#page-61">61</a>, <a href="#page-136">136</a>.</span></li> +<li>Joints of fingers, <span><a href="#page-7">7</a>, <a href="#page-18">18</a>, <a href="#page-173">173</a>.</span></li> +<li>Juri, <span><a href="#page-79">79</a>.</span></li> + + +<li class="newletter">Kamassin, <span><a href="#page-130">130</a>.</span></li> +<li>Kamilaroi, <span><a href="#page-27">27</a>, <a href="#page-107">107</a>, <a href="#page-112">112</a>.</span></li> +<li>Kamtschatka, <span><a href="#page-75">75</a>, <a href="#page-157">157</a>.</span></li> +<li>Kanuri, <span><a href="#page-136">136</a>, <a href="#page-149">149</a>.</span></li> +<li>Karankawa, <span><a href="#page-68">68</a>.</span></li> +<li>Karen, <span><a href="#page-112">112</a>.</span></li> +<li>Keppel Bay, <span><a href="#page-24">24</a>.</span></li> +<li>Ki-Nyassa, <span><a href="#page-150">150</a>.</span></li> +<li>Kiriri, <span><a href="#page-55">55</a>, <a href="#page-138">138</a>, <a href="#page-139">139</a>, <a href="#page-167">167</a>.</span></li> +<li>Kissi, <span><a href="#page-145">145</a>.</span></li> +<li>Ki-Swahili, <span><a href="#page-42">42</a>.</span></li> +<li>Ki-Yau, <span><a href="#page-150">150</a>.</span></li> +<li>Klamath, <span><a href="#page-58">58</a>, <a href="#page-59">59</a>.</span></li> +<li>Knot, <span><a href="#page-7">7</a>, <a href="#page-9">9</a>, <a href="#page-19">19</a>, <a href="#page-40">40</a>, <a href="#page-93">93</a>, <a href="#page-115">115</a>.</span></li> +<li>Kolyma, <span><a href="#page-75">75</a>.</span></li> +<li>Kootenay, <span><a href="#page-65">65</a>.</span></li> +<li>Koriak, <span><a href="#page-75">75</a>.</span></li> +<li>Kredy, <span><a href="#page-149">149</a>.</span></li> +<li>Kru, <span><a href="#page-146">146</a>.</span></li> +<li>Ku-Mbutti, <span><a href="#page-78">78</a>.</span></li> +<li>Kunama, <span><a href="#page-151">151</a>.</span></li> +<li>Kuri, <span><a href="#page-188">188</a>.</span></li> +<li>Kusaie, <span><a href="#page-78">78</a>, <a href="#page-80">80</a>.</span></li> +<li>Kwakiutl, <span><a href="#page-45">45</a>.</span></li> + + +<li class="newletter">Labillardière, <span><a href="#page-85">85</a>.</span></li> +<li>Labrador, <span><a href="#page-29">29</a>.</span></li> +<li>Lake Kopperamana, <span><a href="#page-107">107</a>.</span></li> +<li>Latin, <span><a href="#page-40">40</a>, <a href="#page-44">44</a>, <a href="#page-76">76</a>, <a href="#page-81">81</a>, <a href="#page-86">86</a>, <a href="#page-124">124</a>, <a href="#page-128">128</a>, <a href="#page-168">168</a>, <a href="#page-169">169</a>, <a href="#page-181">181</a>, <a href="#page-182">182</a>.</span></li> +<li>Lazi, <span><a href="#page-189">189</a>.</span></li> +<li>Left hand, <span><a href="#page-10">10–17</a>, <a href="#page-54">54</a>.</span></li> +<li>Leper's Island, <span><a href="#page-16">16</a>.</span></li> +<li>Leptscha, <span><a href="#page-193">193</a>.</span></li> +<li>Lifu, <span><a href="#page-143">143</a>.</span></li> +<li>Little finger, <span><a href="#page-10">10–18</a>, <a href="#page-48">48</a>, <a href="#page-54">54</a>, <a href="#page-61">61</a>, <a href="#page-91">91</a>.</span></li> +<li><a class="pgnum" id="page-216" title="- 216 -"></a>Logone, <span><a href="#page-186">186</a>.</span></li> +<li>London, <span><a href="#page-124">124</a>.</span></li> +<li>Lower California, <span><a href="#page-29">29</a>.</span></li> +<li>Luli, <span><a href="#page-118">118</a>.</span></li> +<li>Lutuami, <span><a href="#page-164">164</a>.</span></li> + + +<li class="newletter">Maba, <span><a href="#page-80">80</a>.</span></li> +<li>Macassar, <span><a href="#page-93">93</a>.</span></li> +<li>Machine, Calculating, <span><a href="#page-19">19</a>, <a href="#page-20">20</a>.</span></li> +<li>Mackenzie River, <span><a href="#page-157">157</a>.</span></li> +<li>Macuni, <span><a href="#page-174">174</a>.</span></li> +<li>Madagascar, <span><a href="#page-8">8</a>, <a href="#page-9">9</a>.</span></li> +<li>Maipures, <span><a href="#page-15">15</a>, <a href="#page-56">56</a>.</span></li> +<li>Mairassis, <span><a href="#page-10">10</a>.</span></li> +<li>Malagasy, <span><a href="#page-83">83</a>, <a href="#page-95">95</a>.</span></li> +<li>Malanta, <span><a href="#page-96">96</a>.</span></li> +<li>Malay, <span><a href="#page-8">8</a>, <a href="#page-45">45</a>, <a href="#page-90">90</a>, <a href="#page-93">93</a>, <a href="#page-170">170</a>.</span></li> +<li>Mallicolo, <span><a href="#page-152">152</a>.</span></li> +<li>Manadu, <span><a href="#page-93">93</a>.</span></li> +<li>Mandingo, <span><a href="#page-186">186</a>.</span></li> +<li>Mangareva, <span><a href="#page-114">114</a>.</span></li> +<li>Manx, <span><a href="#page-180">180</a>.</span></li> +<li>Many, <span><a href="#page-2">2</a>, <a href="#page-21">21–23</a>, <a href="#page-25">25</a>, <a href="#page-28">28</a>, <a href="#page-100">100</a>.</span></li> +<li>Maori, <span><a href="#page-64">64</a>, <a href="#page-93">93</a>, <a href="#page-122">122</a>.</span></li> +<li>Marachowie, <span><a href="#page-26">26</a>.</span></li> +<li>Maré, <span><a href="#page-84">84</a>.</span></li> +<li>Maroura, <span><a href="#page-106">106</a>.</span></li> +<li>Marquesas, <span><a href="#page-93">93</a>, <a href="#page-114">114</a>, <a href="#page-115">115</a>.</span></li> +<li>Marshall Islands, <span><a href="#page-122">122</a>, <a href="#page-152">152</a>.</span></li> +<li>Massachusetts, <span><a href="#page-91">91</a>, <a href="#page-159">159</a>.</span></li> +<li>Mathematician, <span><a href="#page-2">2</a>, <a href="#page-3">3</a>, <a href="#page-35">35</a>, <a href="#page-102">102</a>, <a href="#page-127">127</a>, <a href="#page-210">210</a>.</span></li> +<li>Matibani, <span><a href="#page-151">151</a>.</span></li> +<li>Matlaltzinca, <span><a href="#page-166">166</a>.</span></li> +<li>Maya, <span><a href="#page-45">45</a>, <a href="#page-46">46</a>, <a href="#page-199">199</a>, <a href="#page-205">205</a>.</span></li> +<li>Mbayi, <span><a href="#page-111">111</a>.</span></li> +<li>Mbocobi, <span><a href="#page-22">22</a>.</span></li> +<li>Mbousha, <span><a href="#page-66">66</a>.</span></li> +<li>Melanesia, <span><a href="#page-16">16</a>, <a href="#page-22">22</a>, <a href="#page-28">28</a>, <a href="#page-84">84</a>, <a href="#page-95">95</a>.</span></li> +<li>Mende, <span><a href="#page-186">186</a>.</span></li> +<li>Mexico, <span><a href="#page-29">29</a>, <a href="#page-195">195</a>, <a href="#page-201">201</a>, <a href="#page-204">204</a>, <a href="#page-208">208</a>.</span></li> +<li>Miami, <span><a href="#page-91">91</a>.</span></li> +<li>Micmac, <span><a href="#page-90">90</a>, <a href="#page-160">160</a>.</span></li> +<li>Middle finger, <span><a href="#page-12">12</a>, <a href="#page-15">15</a>, <a href="#page-62">62</a>.</span></li> +<li>Mille, <span><a href="#page-122">122</a>.</span></li> +<li>Minnal Yungar, <span><a href="#page-26">26</a>.</span></li> +<li>Minsi, <span><a href="#page-162">162</a>.</span></li> +<li>Mississaga, <span><a href="#page-44">44</a>, <a href="#page-92">92</a>.</span></li> +<li>Mississippi, <span><a href="#page-125">125</a>.</span></li> +<li>Mocobi, <span><a href="#page-119">119</a>.</span></li> +<li>Mohegan, <span><a href="#page-91">91</a>.</span></li> +<li>Mohican, <span><a href="#page-172">172</a>.</span></li> +<li>Mokko, <span><a href="#page-149">149</a>.</span></li> +<li>Molele, <span><a href="#page-164">164</a>.</span></li> +<li>Moneroo, <span><a href="#page-109">109</a>.</span></li> +<li>Mongolian, <span><a href="#page-8">8</a>.</span></li> +<li>Montagnais, <span><a href="#page-53">53</a>, <a href="#page-54">54</a>, <a href="#page-175">175</a>.</span></li> +<li>Moree, <span><a href="#page-24">24</a>.</span></li> +<li>Moreton Bay, <span><a href="#page-108">108</a>.</span></li> +<li>Mort Noular, <span><a href="#page-107">107</a>.</span></li> +<li>Mosquito, <span><a href="#page-69">69</a>, <a href="#page-70">70</a>, <a href="#page-121">121</a>.</span></li> +<li>Mota, <span><a href="#page-95">95</a>, <a href="#page-153">153</a>.</span></li> +<li>Mpovi, <span><a href="#page-152">152</a>.</span></li> +<li>Multiplication, <span><a href="#page-19">19</a>, <a href="#page-33">33</a>, <a href="#page-40">40</a>, <a href="#page-43">43</a>, <a href="#page-59">59</a>.</span></li> +<li>Mundari, <span><a href="#page-193">193</a>.</span></li> +<li>Mundo, <span><a href="#page-186">186</a>.</span></li> +<li>Muralug, <span><a href="#page-17">17</a>.</span></li> +<li>Murray River, <span><a href="#page-106">106</a>, <a href="#page-109">109</a>.</span></li> +<li>Muysca, <span><a href="#page-206">206</a>.</span></li> + + +<li class="newletter">Nagranda, <span><a href="#page-207">207</a>.</span></li> +<li>Nahuatl, <span><a href="#page-141">141</a>, <a href="#page-144">144</a>, <a href="#page-177">177</a>, <a href="#page-201">201</a>, <a href="#page-205">205</a>.</span></li> +<li>Nakuhiva, <span><a href="#page-116">116</a>, <a href="#page-130">130</a>.</span></li> +<li>Negro, <span><a href="#page-8">8</a>, <a href="#page-9">9</a>, <a href="#page-15">15</a>, <a href="#page-29">29</a>, <a href="#page-184">184</a>.</span></li> +<li>Nengone, <span><a href="#page-63">63</a>, <a href="#page-136">136</a>.</span></li> +<li>New, <span><a href="#page-128">128–130</a>.</span></li> +<li>New Caledonia, <span><a href="#page-154">154</a>.</span></li> +<li>New Granada, <span><a href="#page-195">195</a>.</span></li> +<li>New Guinea, <span><a href="#page-10">10</a>, <a href="#page-152">152</a>.</span></li> +<li>New Hebrides, <span><a href="#page-155">155</a>.</span></li> +<li>New Ireland, <span><a href="#page-45">45</a>.</span></li> +<li>New Zealand, <span><a href="#page-123">123</a>.</span></li> +<li>Nez Perces, <span><a href="#page-65">65</a>, <a href="#page-158">158</a>.</span></li> +<li>Ngarrimowro, <span><a href="#page-110">110</a>.</span></li> +<li>Niam Niam, <span><a href="#page-64">64</a>, <a href="#page-136">136</a>.</span></li> +<li>Nicaragua, <span><a href="#page-80">80</a>.</span></li> +<li>Nicobar, <span><a href="#page-78">78</a>, <a href="#page-193">193</a>.</span></li> +<li>Nightingale, <span><a href="#page-4">4</a>.</span></li> +<li>Nootka, <span><a href="#page-163">163</a>, <a href="#page-198">198</a>.</span></li> +<li>Norman River, <span><a href="#page-24">24</a>.</span></li> +<li>North America, <span><a href="#page-28">28</a>, <a href="#page-82">82</a>, <a href="#page-171">171</a>, <a href="#page-173">173</a>, <a href="#page-176">176</a>, <a href="#page-194">194</a>, <a href="#page-201">201</a>.</span></li> +<li>Notch, <span><a href="#page-7">7</a>, <a href="#page-9">9</a>, <a href="#page-93">93</a>.</span></li> +<li>Numeral frame, <span><a href="#page-19">19</a>.</span></li> +<li>Nupe, <span><a href="#page-149">149</a>, <a href="#page-186">186</a>.</span></li> +<li>Nusqually, <span><a href="#page-96">96</a>.</span></li> + + +<li class="newletter">Oceania, <span><a href="#page-115">115</a>, <a href="#page-176">176</a>.</span></li> +<li>Octonary, <span><a href="#page-100">chap. v</a>.</span></li> +<li>Odessa, <span><a href="#page-34">34</a>.</span></li> +<li>Ojibwa, <span><a href="#page-84">84</a>, <a href="#page-159">159</a>.</span></li> +<li>Okanaken, <span><a href="#page-88">88</a>.</span></li> +<li>Omaha, <span><a href="#page-161">161</a>.</span></li> +<li>Omeo, <span><a href="#page-110">110</a>.</span></li> +<li>Oregon, <span><a href="#page-58">58</a>, <a href="#page-195">195</a>.</span></li> +<li><a class="pgnum" id="page-217" title="- 217 -"></a>Orejone, <span><a href="#page-23">23</a>.</span></li> +<li>Orinoco, <span><a href="#page-54">54</a>, <a href="#page-56">56</a>, <a href="#page-195">195</a>.</span></li> +<li>Ostrich, <span><a href="#page-71">71</a>, <a href="#page-72">72</a>.</span></li> +<li>Otomac, <span><a href="#page-15">15</a>.</span></li> +<li>Otomi, <span><a href="#page-165">165</a>, <a href="#page-199">199</a>.</span></li> +<li>Ottawa, <span><a href="#page-159">159</a>.</span></li> +<li>Oyster Bay, <span><a href="#page-79">79</a>.</span></li> + + +<li class="newletter">Pacific, <span><a href="#page-29">29</a>, <a href="#page-113">113</a>, <a href="#page-116">116</a>, <a href="#page-117">117</a>, <a href="#page-131">131</a>.</span></li> +<li>Palm (of the hand), <span><a href="#page-12">12</a>, <a href="#page-14">14</a>, <a href="#page-15">15</a>.</span></li> +<li>Palm Island, <span><a href="#page-156">156</a>.</span></li> +<li>Pama, <span><a href="#page-136">136</a>, <a href="#page-155">155</a>.</span></li> +<li>Pampanaga, <span><a href="#page-66">66</a>.</span></li> +<li>Papaa, <span><a href="#page-148">148</a>.</span></li> +<li>Paraguay, <span><a href="#page-55">55</a>, <a href="#page-71">71</a>, <a href="#page-118">118</a>, <a href="#page-195">195</a>.</span></li> +<li>Parana, <span><a href="#page-119">119</a>.</span></li> +<li>Paris, <span><a href="#page-182">182</a>.</span></li> +<li>Pawnee, <span><a href="#page-91">91</a>, <a href="#page-121">121</a>, <a href="#page-162">162</a>.</span></li> +<li>Pebble, <span><a href="#page-7">7–9</a>, <a href="#page-19">19</a>, <a href="#page-40">40</a>, <a href="#page-93">93</a>, <a href="#page-179">179</a>.</span></li> +<li>Peno, <span><a href="#page-2">2</a>.</span></li> +<li>Peru (Peruvian), <span><a href="#page-2">2</a>, <a href="#page-22">22</a>, <a href="#page-61">61</a>, <a href="#page-206">206</a>.</span></li> +<li>Philippine, <span><a href="#page-66">66</a>.</span></li> +<li>Philology (Philologist), <span><a href="#page-128">128</a>, <a href="#page-209">209</a>, <a href="#page-210">210</a>.</span></li> +<li>Phœnician, <span><a href="#page-90">90</a>, <a href="#page-179">179</a>.</span></li> +<li>Pigmy, <span><a href="#page-69">69</a>, <a href="#page-70">70</a>, <a href="#page-78">78</a>.</span></li> +<li>Pikumbul, <span><a href="#page-57">57</a>, <a href="#page-138">138</a>.</span></li> +<li>Pines, Isle of, <span><a href="#page-153">153</a>.</span></li> +<li>Pinjarra, <span><a href="#page-26">26</a>.</span></li> +<li>Plenty, <span><a href="#page-25">25</a>, <a href="#page-77">77</a>.</span></li> +<li>Point Barrow, <span><a href="#page-30">30</a>, <a href="#page-51">51</a>, <a href="#page-64">64</a>, <a href="#page-83">83</a>, <a href="#page-137">137</a>, <a href="#page-159">159</a>.</span></li> +<li>Polynesia, <span><a href="#page-22">22</a>, <a href="#page-28">28</a>, <a href="#page-118">118</a>, <a href="#page-130">130</a>, <a href="#page-170">170</a>.</span></li> +<li>Pondo, <span><a href="#page-33">33</a>.</span></li> +<li>Popham Bay, <span><a href="#page-107">107</a>.</span></li> +<li>Port Darwin, <span><a href="#page-109">109</a>.</span></li> +<li>Port Essington, <span><a href="#page-24">24</a>, <a href="#page-107">107</a>.</span></li> +<li>Port Mackay, <span><a href="#page-26">26</a>.</span></li> +<li>Port Macquarie, <span><a href="#page-109">109</a>.</span></li> +<li>Puget Sound, <span><a href="#page-96">96</a>.</span></li> +<li>Puri, <span><a href="#page-22">22</a>, <a href="#page-92">92</a>.</span></li> + + +<li class="newletter">Quappa, <span><a href="#page-171">171</a>, <a href="#page-172">172</a>.</span></li> +<li>Quaternary, <span><a href="#page-100">chap. v</a>.</span></li> +<li>Queanbeyan, <span><a href="#page-24">24</a>.</span></li> +<li>Quiche, <span><a href="#page-205">205</a>.</span></li> +<li>Quichua, <span><a href="#page-61">61</a>.</span></li> + + +<li class="newletter">Rapid, <span><a href="#page-163">163</a>.</span></li> +<li>Rarotonga, <span><a href="#page-114">114</a>.</span></li> +<li>Richmond River, <span><a href="#page-109">109</a>.</span></li> +<li>Right hand, <span><a href="#page-10">10–18</a>, <a href="#page-54">54</a>.</span></li> +<li>Right-handedness, <span><a href="#page-13">13</a>, <a href="#page-14">14</a>.</span></li> +<li>Ring finger, <span><a href="#page-15">15</a>.</span></li> +<li>Rio Grande, <span><a href="#page-195">195</a>.</span></li> +<li>Rio Napo, <span><a href="#page-22">22</a>.</span></li> +<li>Rio Norte, <span><a href="#page-136">136</a>, <a href="#page-199">199</a>.</span></li> +<li>Russia (Russian), <span><a href="#page-30">30</a>, <a href="#page-35">35</a>.</span></li> + + +<li class="newletter">Sahaptin, <span><a href="#page-158">158</a>.</span></li> +<li>San Antonio, <span><a href="#page-136">136</a>.</span></li> +<li>San Blas, <span><a href="#page-79">79</a>, <a href="#page-80">80</a>.</span></li> +<li>Sanskrit, <span><a href="#page-40">40</a>, <a href="#page-92">92</a>, <a href="#page-97">97</a>, <a href="#page-128">128</a>.</span></li> +<li>Sapibocone, <span><a href="#page-84">84</a>, <a href="#page-167">167</a>.</span></li> +<li>Saste (Shasta), <span><a href="#page-165">165</a>.</span></li> +<li>Scratch, <span><a href="#page-7">7</a>.</span></li> +<li>Scythian, <span><a href="#page-169">169</a>.</span></li> +<li>Seed, <span><a href="#page-93">93</a>.</span></li> +<li>Semitic, <span><a href="#page-89">89</a>.</span></li> +<li>Senary, <span><a href="#page-100">chap. v</a>.</span></li> +<li>Sesake, <span><a href="#page-136">136</a>, <a href="#page-155">155</a>.</span></li> +<li>Several, <span><a href="#page-22">22</a>.</span></li> +<li>Sexagesimal, <span><a href="#page-124">124</a>, <a href="#page-208">208</a>.</span></li> +<li>Shawnoe, <span><a href="#page-160">160</a>.</span></li> +<li>Shell, <span><a href="#page-7">7</a>, <a href="#page-19">19</a>, <a href="#page-70">70</a>, <a href="#page-93">93</a>.</span></li> +<li>Shushwap, <span><a href="#page-88">88</a>.</span></li> +<li>Siberia, <span><a href="#page-29">29</a>, <a href="#page-30">30</a>, <a href="#page-187">187</a>, <a href="#page-190">190</a>.</span></li> +<li>Sierra Leone, <span><a href="#page-83">83</a>.</span></li> +<li>Sign language, <span><a href="#page-6">6</a>.</span></li> +<li>Sioux, <span><a href="#page-83">83</a>.</span></li> +<li>Slang, <span><a href="#page-124">124</a>.</span></li> +<li>Slavonic, <span><a href="#page-40">40</a>.</span></li> +<li>Snowy River, <span><a href="#page-110">110</a>.</span></li> +<li>Soussou, <span><a href="#page-83">83</a>, <a href="#page-147">147</a>.</span></li> +<li>South Africa, <span><a href="#page-4">4</a>, <a href="#page-15">15</a>, <a href="#page-28">28</a>.</span></li> +<li>South America, <span><a href="#page-2">2</a>, <a href="#page-15">15</a>, <a href="#page-22">22</a>, <a href="#page-23">23</a>, <a href="#page-27">27–29</a>, <a href="#page-54">54</a>, <a href="#page-57">57</a>, <a href="#page-72">72</a>, <a href="#page-76">76</a>, <a href="#page-78">78</a>, <a href="#page-79">79</a>, <a href="#page-104">104</a>, <a href="#page-110">110</a>, <a href="#page-173">173</a>, <a href="#page-174">174</a>, <a href="#page-194">194</a>, <a href="#page-201">201</a>, <a href="#page-206">206</a>.</span></li> +<li>Spanish, <span><a href="#page-2">2</a>, <a href="#page-23">23</a>, <a href="#page-42">42</a>.</span></li> +<li>Splint, <span><a href="#page-7">7</a>.</span></li> +<li>Stick, <span><a href="#page-7">7</a>, <a href="#page-179">179</a>.</span></li> +<li>Stlatlumh, <span><a href="#page-88">88</a>.</span></li> +<li>Streaky Bay, <span><a href="#page-26">26</a>.</span></li> +<li>String, <span><a href="#page-7">7</a>, <a href="#page-9">9</a>, <a href="#page-64">64</a>, <a href="#page-71">71</a>.</span></li> +<li>Strong's Island, <span><a href="#page-78">78</a>.</span></li> +<li>Subtraction, <span><a href="#page-19">19</a>, <a href="#page-44">44–47</a>.</span></li> +<li>Sunda, <span><a href="#page-120">120</a>.</span></li> +<li>Sweden (Swedish), <span><a href="#page-129">129</a>, <a href="#page-132">132</a>, <a href="#page-183">183</a>.</span></li> + + +<li class="newletter">Tacona, <span><a href="#page-2">2</a>.</span></li> +<li>Taensa, <span><a href="#page-44">44</a>.</span></li> +<li>Tagala, <span><a href="#page-66">66</a>.</span></li> +<li>Tahiti, <span><a href="#page-114">114</a>.</span></li> +<li>Tahuata, <span><a href="#page-115">115</a>.</span></li> +<li>Tama, <span><a href="#page-111">111</a>.</span></li> +<li>Tamanac, <span><a href="#page-54">54</a>, <a href="#page-135">135</a>.</span></li> +<li><a class="pgnum" id="page-218" title="- 218 -"></a>Tambi, <span><a href="#page-120">120</a>.</span></li> +<li>Tanna, <span><a href="#page-154">154</a>.</span></li> +<li>Tarascan, <span><a href="#page-165">165</a>.</span></li> +<li>Tariana, <span><a href="#page-174">174</a>.</span></li> +<li>Tasmania, <span><a href="#page-24">24</a>, <a href="#page-27">27</a>, <a href="#page-79">79</a>, <a href="#page-104">104</a>, <a href="#page-106">106</a>.</span></li> +<li>Tawgy, <span><a href="#page-130">130</a>.</span></li> +<li>Tchetchnia, <span><a href="#page-188">188</a>.</span></li> +<li>Tchiglit, <span><a href="#page-157">157</a>, <a href="#page-196">196</a>.</span></li> +<li>Tembu, <span><a href="#page-33">33</a>.</span></li> +<li>Temne, <span><a href="#page-148">148</a>.</span></li> +<li>Ternary, <span><a href="#page-100">chap. v</a>.</span></li> +<li>Terraba, <span><a href="#page-172">172</a>.</span></li> +<li>Teutonic, <span><a href="#page-40">40</a>, <a href="#page-41">41</a>, <a href="#page-43">43</a>, <a href="#page-179">179</a>, <a href="#page-181">181</a>, <a href="#page-208">208</a>.</span></li> +<li>Texas, <span><a href="#page-69">69</a>.</span></li> +<li>Thibet, <span><a href="#page-96">96</a>.</span></li> +<li>Thumb, <span><a href="#page-10">10–18</a>, <a href="#page-54">54</a>, <a href="#page-59">59</a>, <a href="#page-61">61</a>, <a href="#page-62">62</a>, <a href="#page-113">113</a>, <a href="#page-173">173</a>.</span></li> +<li>Thusch, <span><a href="#page-189">189</a>.</span></li> +<li>Ticuna, <span><a href="#page-168">168</a>.</span></li> +<li>Timukua, <span><a href="#page-165">165</a>.</span></li> +<li>Tlingit, <span><a href="#page-136">136</a>, <a href="#page-163">163</a>, <a href="#page-197">197</a>.</span></li> +<li>Tobi, <span><a href="#page-156">156</a>.</span></li> +<li>Tonga, <span><a href="#page-33">33</a>, <a href="#page-85">85</a>.</span></li> +<li>Torres, <span><a href="#page-17">17</a>, <a href="#page-96">96</a>, <a href="#page-104">104</a>, <a href="#page-105">105</a>.</span></li> +<li>Totonaco, <span><a href="#page-205">205</a>.</span></li> +<li>Towka, <span><a href="#page-78">78</a>.</span></li> +<li>Triton's Bay, <span><a href="#page-152">152</a>.</span></li> +<li>Tschukshi, <span><a href="#page-156">156</a>, <a href="#page-191">191</a>.</span></li> +<li>Tsimshian, <span><a href="#page-86">86</a>, <a href="#page-164">164</a>, <a href="#page-198">198</a>.</span></li> +<li>Tweed River, <span><a href="#page-26">26</a>.</span></li> + + +<li class="newletter">Uainuma, <span><a href="#page-122">122</a>.</span></li> +<li>Udi, <span><a href="#page-188">188</a>.</span></li> +<li>Uea, <span><a href="#page-67">67</a>, <a href="#page-153">153</a>.</span></li> +<li>United States, <span><a href="#page-29">29</a>, <a href="#page-83">83</a>, <a href="#page-195">195</a>.</span></li> +<li>Upper Yarra, <span><a href="#page-110">110</a>.</span></li> +<li>Ureparapara, <span><a href="#page-153">153</a>.</span></li> + + +<li class="newletter">Vaturana, <span><a href="#page-96">96</a>.</span></li> +<li>Vedda, <span><a href="#page-28">28</a>, <a href="#page-31">31</a>, <a href="#page-76">76</a>, <a href="#page-174">174</a>.</span></li> +<li>Vei, <span><a href="#page-16">16</a>, <a href="#page-147">147</a>, <a href="#page-185">185</a>.</span></li> +<li>Victoria, <span><a href="#page-156">156</a>.</span></li> +<li>Vilelo, <span><a href="#page-60">60</a>.</span></li> + + +<li class="newletter">Waiclatpu, <span><a href="#page-164">164</a>.</span></li> +<li>Wales (Welsh), <span><a href="#page-35">35</a>, <a href="#page-46">46</a>, <a href="#page-141">141</a>, <a href="#page-144">144</a>, <a href="#page-177">177</a>, <a href="#page-180">180</a>.</span></li> +<li>Wallachia, <span><a href="#page-121">121</a>.</span></li> +<li>Warrego, <span><a href="#page-107">107</a>, <a href="#page-109">109</a>.</span></li> +<li>Warrior Island, <span><a href="#page-107">107</a>.</span></li> +<li>Wasp, <span><a href="#page-5">5</a>.</span></li> +<li>Watchandie, <span><a href="#page-29">29</a>, <a href="#page-77">77</a>.</span></li> +<li>Watji, <span><a href="#page-120">120</a>.</span></li> +<li>Weedookarry, <span><a href="#page-24">24</a>.</span></li> +<li>Wimmera, <span><a href="#page-107">107</a>.</span></li> +<li>Winnebago, <span><a href="#page-85">85</a>.</span></li> +<li>Wiraduroi, <span><a href="#page-27">27</a>, <a href="#page-108">108</a>.</span></li> +<li>Wirri-Wirri, <span><a href="#page-108">108</a>.</span></li> +<li>Wokke, <span><a href="#page-112">112</a>.</span></li> +<li>Worcester, Mass., Schools of, <span><a href="#page-11">11</a>.</span></li> + + +<li class="newletter">Yahua, <span><a href="#page-168">168</a>.</span></li> +<li>Yaruro, <span><a href="#page-139">139</a>.</span></li> +<li>Yengen, <span><a href="#page-154">154</a>.</span></li> +<li>Yit-tha, <span><a href="#page-109">109</a>.</span></li> +<li>Yoruba, <span><a href="#page-33">33</a>, <a href="#page-47">47</a>, <a href="#page-64">64</a>, <a href="#page-70">70</a>, <a href="#page-185">185</a>.</span></li> +<li>Yucatan, <span><a href="#page-195">195</a>, <a href="#page-201">201</a>.</span></li> +<li>Yuckaburra, <span><a href="#page-26">26</a>.</span></li> + + +<li class="newletter">Zamuco, <span><a href="#page-55">55</a>, <a href="#page-60">60</a>, <a href="#page-138">138</a>, <a href="#page-139">139</a>.</span></li> +<li>Zapara, <span><a href="#page-111">111</a>.</span></li> +<li>Zulu, <span><a href="#page-16">16</a>, <a href="#page-62">62</a>.</span></li> +<li>Zuñi, <span><a href="#page-13">13</a>, <a href="#page-14">14</a>, <a href="#page-48">48</a>, <a href="#page-49">49</a>, <a href="#page-53">53</a>, <a href="#page-54">54</a>, <a href="#page-60">60</a>, <a href="#page-83">83</a>, <a href="#page-137">137</a>.</span></li> +</ul> + + + + +<hr class="front"/> + +<h2><a class="pgnum" id="page-219" title="- 219 -"></a>Footnotes:</h2> + +<div class="footnotes"> + +<p><a id="FN-1" href="#FNA-1"><sup>1</sup></a>Brinton, D. G., <i>Essays of an Americanist</i>, p. 406; and <i>American +Race</i>, p. 359.</p> + +<p><a id="FN-2" href="#FNA-2"><sup>2</sup></a>This information I received from Dr. Brinton by letter.</p> + +<p><a id="FN-3" href="#FNA-3"><sup>3</sup></a>Tylor, <i>Primitive Culture</i>, Vol. I. p. 240.</p> + +<p><a id="FN-4" href="#FNA-4"><sup>4</sup></a><i>Nature</i>, Vol. XXXIII. p. 45.</p> + +<p><a id="FN-5" href="#FNA-5"><sup>5</sup></a>Spix and Martius, <i>Travels in Brazil</i>, Tr. from German by +H. E. Lloyd, Vol. II. p. 255.</p> + +<p><a id="FN-6" href="#FNA-6"><sup>6</sup></a>De Flacourt, <i>Histoire de le grande Isle de Madagascar</i>, ch. xxviii. +Quoted by Peacock, <i>Encyc. Met.</i>, Vol. I. p. 393.</p> + +<p><a id="FN-7" href="#FNA-7"><sup>7</sup></a>Bellamy, Elizabeth W., <i>Atlantic Monthly</i>, March, 1893, p. 317.</p> + +<p><a id="FN-8" href="#FNA-8"><sup>8</sup></a><i>Grundriss der Sprachwissenschaft</i>, Bd. III. Abt. i., p. 94.</p> + +<p><a id="FN-9" href="#FNA-9"><sup>9</sup></a>Pruner-Bey, <i>Bulletin de la Société d'Anthr. de Paris</i>, 1861, p. 462.</p> + +<p><a id="FN-10" href="#FNA-10"><sup>10</sup></a>“Manual Concepts,” <i>Am. Anthropologist</i>, 1892, p. 292.</p> + +<p><a id="FN-11" href="#FNA-11"><sup>11</sup></a>Tylor, <i>Primitive Culture</i>, Vol. I. p. 245.</p> + +<p><a id="FN-12" href="#FNA-12"><sup>12</sup></a><i>Op. cit.</i>, <i>loc. cit.</i></p> + +<p><a id="FN-13" href="#FNA-13"><sup>13</sup></a>“Aboriginal Inhabitants of Andaman Islands,” <i>Journ. Anth. Inst.</i>, +1882, p. 100.</p> + +<p><a id="FN-14" href="#FNA-14"><sup>14</sup></a>Morice, A., <i>Revue d'Anthropologie</i>, 1878, p. 634.</p> + +<p><a id="FN-15" href="#FNA-15"><sup>15</sup></a>Macdonald, J., “Manners, Customs, etc., of South African Tribes,” +<i>Journ. Anthr. Inst.</i>, 1889, p. 290. About a dozen tribes are enumerated +by Mr. Macdonald: Pondos, Tembucs, Bacas, Tolas, etc.</p> + +<p><a id="FN-16" href="#FNA-16"><sup>16</sup></a>Codrington, R. H., <i>Melanesians, their Anthropology and Folk-Lore</i>, p. 353.</p> + +<p><a id="FN-17" href="#FNA-17"><sup>17</sup></a><i>E.g.</i> the Zuñis. See Cushing's paper quoted above.</p> + +<p><a id="FN-18" href="#FNA-18"><sup>18</sup></a>Haddon, A. C., “Ethnography Western Tribes Torres Strait,” <i>Journ. +Anth. Inst.</i>, 1889, p. 305. For a similar method, see <i>Life in the Southern +Isles</i>, by W. W. Gill.</p> + +<p><a id="FN-19" href="#FNA-19"><sup>19</sup></a>Tylor, <i>Primitive Culture</i>, Vol. I. p. 246.</p> + +<p><a id="FN-20" href="#FNA-20"><sup>20</sup></a>Brinton, D. G., Letter of Sept. 23, 1893.</p> + +<p><a id="FN-21" href="#FNA-21"><sup>21</sup></a><i>Ibid</i>. The reference for the Mbocobi, <i>infra</i>, is the same. See also +Brinton's <i>American Race</i>, p. 361.</p> + +<p><a id="FN-22" href="#FNA-22"><sup>22</sup></a>Tylor, <i>Primitive Culture</i>, Vol. I. p. 243.</p> + +<p><a id="FN-23" href="#FNA-23"><sup>23</sup></a><i>Op. cit.</i>, <i>loc. cit.</i></p> + +<p><a id="FN-24" href="#FNA-24"><sup>24</sup></a>Hyades, <i>Bulletin de la Société d'Anthr. de Paris</i>, 1887, p. 340.</p> + +<p><a id="FN-25" href="#FNA-25"><sup>25</sup></a>Wiener, C., <i>Pérou et Bolivie</i>, p. 360.</p> + +<p><a id="FN-26" href="#FNA-26"><sup>26</sup></a>Marcoy, P., <i>Travels in South America</i>, Vol. II p. 47. According to +the same authority, most of the tribes of the Upper Amazon cannot +count above 2 or 3 except by reduplication.</p> + +<p><a id="FN-27" href="#FNA-27"><sup>27</sup></a><i>Op. cit.</i>, Vol. II. p. 281.</p> + +<p><a id="FN-28" href="#FNA-28"><sup>28</sup></a><i>Glossaria Linguarum Brasiliensium</i>. Bororos, p. 15; Guachi, p. +133; Carajas, p. 265.</p> + +<p><a id="FN-29" href="#FNA-29"><sup>29</sup></a>Curr, E. M., <i>The Australian Race</i>, Vol. I. p. 282. The next eight +lists are, in order, from I. p. 294, III. p. 424, III. p. 114, III. p. 124, II. +p. 344, II. p. 308, I. p. 314, III. p. 314, respectively.</p> + +<p><a id="FN-30" href="#FNA-30"><sup>30</sup></a>Bonwick, J., <i>The Daily Life and Origin of the Tasmanians</i>, p. 144.</p> + +<p><a id="FN-31" href="#FNA-31"><sup>31</sup></a>Latham, <i>Comparative Philology</i>, p. 336.</p> + +<p><a id="FN-32" href="#FNA-32"><sup>32</sup></a><i>The Australian Race</i>, Vol. I. p. 205.</p> + +<p><a id="FN-33" href="#FNA-33"><sup>33</sup></a>Mackenzie, A., “Native Australian Langs.,” <i>Journ. Anthr. Inst.</i>, 1874, +p. 263.</p> + +<p><a id="FN-34" href="#FNA-34"><sup>34</sup></a>Curr, <i>The Australian Race</i>, Vol. II. p. 134. The next four lists are +from II. p. 4, I. p. 322, I. p. 346, and I. p. 398, respectively.</p> + +<p><a id="FN-35" href="#FNA-35"><sup>35</sup></a>Curr, <i>op. cit.</i>, Vol. III. p. 50.</p> + +<p><a id="FN-36" href="#FNA-36"><sup>36</sup></a><i>Op. cit.</i>, Vol. III. p. 236.</p> + +<p><a id="FN-37" href="#FNA-37"><sup>37</sup></a>Müller, <i>Sprachwissenschaft</i>. II. i. p. 23.</p> + +<p><a id="FN-38" href="#FNA-38"><sup>38</sup></a><i>Op. cit.</i>, II. i. p. 31.</p> + +<p><a id="FN-39" href="#FNA-39"><sup>39</sup></a>Bonwick, <i>op. cit.</i>, p. 143.</p> + +<p><a id="FN-40" href="#FNA-40"><sup>40</sup></a>Curr, <i>op. cit.</i>, Vol. I. p. 31.</p> + +<p><a id="FN-41" href="#FNA-41"><sup>41</sup></a>Deschamps, <i>L'Anthropologie</i>, 1891, p. 318.</p> + +<p><a id="FN-42" href="#FNA-42"><sup>42</sup></a>Man, E. H. <i>Aboriginal Inhabitants of the Andaman Islands</i>, p. 32.</p> + +<p><a id="FN-43" href="#FNA-43"><sup>43</sup></a>Müller, <i>Sprachwissenschaft</i>, I. ii. p. 29.</p> + +<p><a id="FN-44" href="#FNA-44"><sup>44</sup></a>Oldfield, A., Tr. Eth. Soc. Vol. III. p. 291.</p> + +<p><a id="FN-45" href="#FNA-45"><sup>45</sup></a>Bancroft, H. H., <i>Native Races</i>, Vol. I. p. 564.</p> + +<p><a id="FN-46" href="#FNA-46"><sup>46</sup></a>“Notes on Counting, etc., among the Eskimos of Point Barrow.” +<i>Am. Anthrop.</i>, 1890, p. 38.</p> + +<p><a id="FN-47" href="#FNA-47"><sup>47</sup></a><i>Second Voyage</i>, p. 556.</p> + +<p><a id="FN-48" href="#FNA-48"><sup>48</sup></a><i>Personal Narrative</i>, Vol. I. p. 311.</p> + +<p><a id="FN-49" href="#FNA-49"><sup>49</sup></a>Burton, B. F., <i>Mem. Anthr. Soc. of London</i>, Vol. I. p. 314.</p> + +<p><a id="FN-50" href="#FNA-50"><sup>50</sup></a><i>Confessions</i>. In collected works, Edinburgh, 1890, Vol. III. p. 337.</p> + +<p><a id="FN-51" href="#FNA-51"><sup>51</sup></a>Ellis, Robert, <i>On Numerals as Signs of Primeval Unity</i>. See also +<i>Peruvia Scythia</i>, by the same author.</p> + +<p><a id="FN-52" href="#FNA-52"><sup>52</sup></a>Stanley, H. M., <i>In Darkest Africa</i>, Vol. II. p. 493.</p> + +<p><a id="FN-53" href="#FNA-53"><sup>53</sup></a>Stanley, H. M., <i>Through the Dark Continent</i>, Vol. II. p. 486.</p> + +<p><a id="FN-54" href="#FNA-54"><sup>54</sup></a>Haumontè, Parisot, Adam, <i>Grammaire et Vocabulaire de la Langue +Taensa</i>, p. 20.</p> + +<p><a id="FN-55" href="#FNA-55"><sup>55</sup></a>Chamberlain, A. F., <i>Lang. of the Mississaga Indians of Skugog. Vocab.</i></p> + +<p><a id="FN-56" href="#FNA-56"><sup>56</sup></a>Boas, Fr., <i>Sixth Report on the Indians of the Northwest</i>, p. 105.</p> + +<p><a id="FN-57" href="#FNA-57"><sup>57</sup></a>Beauregard, O., <i>Bulletin de la Soc. d'Anthr. de Paris</i>, 1886, p. 526.</p> + +<p><a id="FN-58" href="#FNA-58"><sup>58</sup></a>Ray, S. H., <i>Journ. Anthr. Inst.</i>, 1891, p. 8.</p> + +<p><a id="FN-59" href="#FNA-59"><sup>59</sup></a><i>Op. cit.</i>, p. 12.</p> + +<p><a id="FN-60" href="#FNA-60"><sup>60</sup></a>Müller, <i>Sprachwissenschaft</i>, IV. i. p. 136.</p> + +<p><a id="FN-61" href="#FNA-61"><sup>61</sup></a>Brinton, <i>The Maya Chronicles</i>, p. 50.</p> + +<p><a id="FN-62" href="#FNA-62"><sup>62</sup></a>Trumbull, <i>On Numerals in Am. Ind. Lang.</i>, p. 35.</p> + +<p><a id="FN-63" href="#FNA-63"><sup>63</sup></a>Boas, Fr. This information was received directly from Dr. Boas. +It has never before been published.</p> + +<p><a id="FN-64" href="#FNA-64"><sup>64</sup></a>Bancroft, H. H., <i>Native Races</i>, Vol. II. p. 753. See also <a href="#page-199">p. 199</a>, <i>infra</i>.</p> + +<p><a id="FN-65" href="#FNA-65"><sup>65</sup></a>Mann, A., “Notes on the Numeral Syst. of the Yoruba Nation,” +<i>Journ. Anth. Inst.</i>, 1886, p. 59, <i>et seq.</i></p> + +<p><a id="FN-66" href="#FNA-66"><sup>66</sup></a>Müller, <i>Sprachwissenschaft</i>, IV. i. p. 202.</p> + +<p><a id="FN-67" href="#FNA-67"><sup>67</sup></a>Trumbull, J. H., <i>On Numerals in Am. Ind. Langs.</i>, p. 11.</p> + +<p><a id="FN-68" href="#FNA-68"><sup>68</sup></a>Cushing, F. H., “Manual Concepts,” <i>Am. Anthr.</i>, 1892, p. 289.</p> + +<p><a id="FN-69" href="#FNA-69"><sup>69</sup></a>Grimm, <i>Geschichte der deutschen Sprache</i>, Vol. I. p. 239.</p> + +<p><a id="FN-70" href="#FNA-70"><sup>70</sup></a>Murdoch, J., <i>American Anthropologist</i>, 1890, p. 39.</p> + +<p><a id="FN-71" href="#FNA-71"><sup>71</sup></a>Kleinschmidt, S., <i>Grammatik der Grönlandischen Sprache</i>, p. 37.</p> + +<p><a id="FN-72" href="#FNA-72"><sup>72</sup></a>Brinton, <i>The Arawak Lang. of Guiana</i>, p. 4.</p> + +<p><a id="FN-73" href="#FNA-73"><sup>73</sup></a>Petitot, E., <i>Dictionnaire de la langue Dènè-Dindjie</i>, p. lv.</p> + +<p><a id="FN-74" href="#FNA-74"><sup>74</sup></a>Gilij, F. S., <i>Saggio di Storia Am.</i>, Vol. II. p. 333.</p> + +<p><a id="FN-75" href="#FNA-75"><sup>75</sup></a>Müller, <i>Sprachwissenschaft</i>, II. i. p. 389.</p> + +<p><a id="FN-76" href="#FNA-76"><sup>76</sup></a><i>Op. cit.</i>, p. 395.</p> + +<p><a id="FN-77" href="#FNA-77"><sup>77</sup></a>Müller, <i>Sprachwissenschaft</i>, II. i. p. 438.</p> + +<p><a id="FN-78" href="#FNA-78"><sup>78</sup></a>Peacock, “Arithmetic,” in <i>Encyc. Metropolitana</i>, 1, p. 480.</p> + +<p><a id="FN-79" href="#FNA-79"><sup>79</sup></a>Brinton, D. G., “The Betoya Dialects,” <i>Proc. Am. Philos. Soc.</i>, 1892, +p. 273.</p> + +<p><a id="FN-80" href="#FNA-80"><sup>80</sup></a>Ridley, W., “Report on Australian Languages and Traditions.” +<i>Journ. Anth. Inst.</i>, 1873, p. 262.</p> + +<p><a id="FN-81" href="#FNA-81"><sup>81</sup></a>Gatschet, “Gram. Klamath Lang.” <i>U. S. Geog. and Geol. Survey</i>, Vol. II. part 1, pp. 524 and 536.</p> + +<p><a id="FN-82" href="#FNA-82"><sup>82</sup></a>Letter of Nov. 17, 1893.</p> + +<p><a id="FN-83" href="#FNA-83"><sup>83</sup></a>Müller, <i>Sprachwissenschaft</i>, II. i. p. 439.</p> + +<p><a id="FN-84" href="#FNA-84"><sup>84</sup></a>Hale, “Indians of No. West. Am.,” <i>Tr. Am. Eth. Soc.</i>, Vol. II. p. 82.</p> + +<p><a id="FN-85" href="#FNA-85"><sup>85</sup></a>Brinton, D. G., <i>Studies in So. Am. Native Languages</i>, p. 25.</p> + +<p><a id="FN-86" href="#FNA-86"><sup>86</sup></a><i>Tr. Am. Philological Association</i>, 1874, p. 41.</p> + +<p><a id="FN-87" href="#FNA-87"><sup>87</sup></a>Tylor, <i>Primitive Culture</i>, Vol. I. p. 251.</p> + +<p><a id="FN-88" href="#FNA-88"><sup>88</sup></a>Müller, <i>Sprachwissenschaft</i>, IV. i. p. 27.</p> + +<p><a id="FN-89" href="#FNA-89"><sup>89</sup></a>See <i>infra</i>, Chapter VII.</p> + +<p><a id="FN-90" href="#FNA-90"><sup>90</sup></a>Ellis, A. B., <i>Ewe Speaking Peoples</i>, etc., p. 253.</p> + +<p><a id="FN-91" href="#FNA-91"><sup>91</sup></a>Tylor, <i>Primitive Culture</i>, Vol. I. p. 256.</p> + +<p><a id="FN-92" href="#FNA-92"><sup>92</sup></a>Stanley, <i>In Darkest Africa</i>, Vol. II. p. 493.</p> + +<p><a id="FN-93" href="#FNA-93"><sup>93</sup></a>Chamberlain, A. F., <i>Proc. Brit. Ass. Adv. of Sci.</i>, 1892, p. 599.</p> + +<p><a id="FN-94" href="#FNA-94"><sup>94</sup></a>Boas, Fr., “Sixth Report on Northwestern Tribes of Canada,” <i>Proc. +Brit. Ass. Adv. Sci.</i>, 1890, p. 657.</p> + +<p><a id="FN-95" href="#FNA-95"><sup>95</sup></a>Hale, H., “Indians of Northwestern Am.,” <i>Tr. Am. Eth. Soc.</i>, Vol. II. +p. 88.</p> + +<p><a id="FN-96" href="#FNA-96"><sup>96</sup></a><i>Op. cit.</i>, p. 95.</p> + +<p><a id="FN-97" href="#FNA-97"><sup>97</sup></a>Müller, <i>Sprachwissenschaft</i>, II. ii. p. 147.</p> + +<p><a id="FN-98" href="#FNA-98"><sup>98</sup></a>Schoolcraft, <i>Archives of Aboriginal Knowledge</i>, Vol. IV. p. 429.</p> + +<p><a id="FN-99" href="#FNA-99"><sup>99</sup></a>Du Chaillu, P. B., <i>Tr. Eth. Soc.</i>, London, Vol. I. p. 315.</p> + +<p><a id="FN-100" href="#FNA-100"><sup>100</sup></a>Latham, R. G., <i>Essays, chiefly Philological and Ethnographical</i>, p. +247. The above are so unlike anything else in the world, that they are +not to be accepted without careful verification.</p> + +<p><a id="FN-101" href="#FNA-101"><sup>101</sup></a>Pott, <i>Zählmethode</i>, p. 45.</p> + +<p><a id="FN-102" href="#FNA-102"><sup>102</sup></a>Gatschet, A. S., <i>The Karankawa Indians, the Coast People of +Texas</i>. The meanings of 6, 7, 8, and 9 are conjectural with me.</p> + +<p><a id="FN-103" href="#FNA-103"><sup>103</sup></a>Stanley, H. M., <i>In Darkest Africa</i>, Vol. II. p. 492.</p> + +<p><a id="FN-104" href="#FNA-104"><sup>104</sup></a>Müller, <i>Sprachwissenschaft</i>, II. i. p. 317.</p> + +<p><a id="FN-105" href="#FNA-105"><sup>105</sup></a>Toy, C. H., <i>Trans. Am. Phil. Assn.</i>, 1878, p. 29.</p> + +<p><a id="FN-106" href="#FNA-106"><sup>106</sup></a>Burton, R. F., <i>Mem. Anthrop. Soc. of London</i>. 1, p. 314. In the +illustration which follows, Burton gives 6820, instead of 4820; which is +obviously a misprint.</p> + +<p><a id="FN-107" href="#FNA-107"><sup>107</sup></a>Dobrizhoffer, <i>History of the Abipones</i>, Vol. II. p. 169.</p> + +<p><a id="FN-108" href="#FNA-108"><sup>108</sup></a>Sayce, A. H., <i>Comparative Philology</i>, p. 254.</p> + +<p><a id="FN-109" href="#FNA-109"><sup>109</sup></a><i>Tr. Eth. Society of London </i>, Vol. III. p. 291.</p> + +<p><a id="FN-110" href="#FNA-110"><sup>110</sup></a>Ray, S. H., <i>Journ. Anthr. Inst.</i>, 1889, p. 501.</p> + +<p><a id="FN-111" href="#FNA-111"><sup>111</sup></a>Stanley, <i>In Darkest Africa</i>, Vol. II. p. 492.</p> + +<p><a id="FN-112" href="#FNA-112"><sup>112</sup></a><i>Op. cit.</i>, <i>loc. cit.</i></p> + +<p><a id="FN-113" href="#FNA-113"><sup>113</sup></a>Tylor, <i>Primitive Culture</i>, Vol. I. p. 249.</p> + +<p><a id="FN-114" href="#FNA-114"><sup>114</sup></a>Müller, <i>Sprachwissenschaft</i>, IV. i. p. 36.</p> + +<p><a id="FN-115" href="#FNA-115"><sup>115</sup></a>Martius, <i>Glos. Ling. Brasil.</i>, p. 271.</p> + +<p><a id="FN-116" href="#FNA-116"><sup>116</sup></a>Tylor, <i>Primitive Culture</i>, Vol. I. p. 248.</p> + +<p><a id="FN-117" href="#FNA-117"><sup>117</sup></a>Roth, H. Ling, <i>Aborigines of Tasmania</i>, p. 146.</p> + +<p><a id="FN-118" href="#FNA-118"><sup>118</sup></a>Lull, E. P., <i>Tr. Am. Phil, Soc.</i>, 1873, p. 108.</p> + +<p><a id="FN-119" href="#FNA-119"><sup>119</sup></a>Ray, S. H. “Sketch of Api Gram.,” <i>Journ. Anthr. Inst.</i>, 1888, p. 300.</p> + +<p><a id="FN-120" href="#FNA-120"><sup>120</sup></a>Kleinschmidt, S., <i>Grammatik der Grönlandischen Spr.</i>, p. 39.</p> + +<p><a id="FN-121" href="#FNA-121"><sup>121</sup></a>Müller, <i>Sprachwissenschaft</i>, I. ii. p. 184.</p> + +<p><a id="FN-122" href="#FNA-122"><sup>122</sup></a><i>Op. cit.</i>, I. ii. p. 18, and II. i. p. 222.</p> + +<p><a id="FN-123" href="#FNA-123"><sup>123</sup></a>Squier, G. E., <i>Nicaragua</i>, Vol. II. p. 326.</p> + +<p><a id="FN-124" href="#FNA-124"><sup>124</sup></a>Schoolcraft, H. R., <i>Archives of Aboriginal Knowledge</i>, Vol. II. p. 208.</p> + +<p><a id="FN-125" href="#FNA-125"><sup>125</sup></a>Tylor, <i>Primitive Culture</i>, Vol. I. p. 264.</p> + +<p><a id="FN-126" href="#FNA-126"><sup>126</sup></a>Goedel, “Ethnol. des Soussous,” <i>Bull. de la Soc. d'Anthr. de Paris</i>, +1892, p. 185.</p> + +<p><a id="FN-127" href="#FNA-127"><sup>127</sup></a>Ellis, W., <i>History of Madagascar</i>, Vol. I. p. 507.</p> + +<p><a id="FN-128" href="#FNA-128"><sup>128</sup></a>Beauregard, O., <i>Bull. de la Soc. d'Anthr. de Paris</i>, 1886, p. 236.</p> + +<p><a id="FN-129" href="#FNA-129"><sup>129</sup></a>Schoolcraft, H. R., <i>Archives of Aboriginal Knowledge</i>, Vol. II. p. 207.</p> + +<p><a id="FN-130" href="#FNA-130"><sup>130</sup></a>Tylor, <i>Primitive Culture</i>, Vol. I. p. 249.</p> + +<p><a id="FN-131" href="#FNA-131"><sup>131</sup></a><i>Op. cit.</i> Vol. I. p. 250.</p> + +<p><a id="FN-132" href="#FNA-132"><sup>132</sup></a>Peacock, <i>Encyc. Metropolitana</i>, 1, p. 478.</p> + +<p><a id="FN-133" href="#FNA-133"><sup>133</sup></a><i>Op. cit.</i>, <i>loc. cit.</i></p> + +<p><a id="FN-134" href="#FNA-134"><sup>134</sup></a>Schoolcraft, H. R., <i>Archives of Aboriginal Knowledge</i>, Vol. II. p. 213.</p> + +<p><a id="FN-135" href="#FNA-135"><sup>135</sup></a><i>Op. cit.</i>, p. 216.</p> + +<p><a id="FN-136" href="#FNA-136"><sup>136</sup></a><i>Op. cit.</i>, p. 206.</p> + +<p><a id="FN-137" href="#FNA-137"><sup>137</sup></a>Mariner, <i>Gram. Tonga Lang.</i>, last part of book. [Not paged.]</p> + +<p><a id="FN-138" href="#FNA-138"><sup>138</sup></a>Morice, A. G., “The Déné Langs,” <i>Trans. Can. Inst.</i>, March 1890, +p. 186.</p> + +<p><a id="FN-139" href="#FNA-139"><sup>139</sup></a>Boas, Fr., “Fifth Report on the Northwestern Tribes of Canada,” +<i>Proc. Brit. Ass. Adv. of Science</i>, 1889, p. 881.</p> + +<p><a id="FN-140" href="#FNA-140"><sup>140</sup></a><i>Do. Sixth Rep.</i>, 1890, pp. 684, 686, 687.</p> + +<p><a id="FN-141" href="#FNA-141"><sup>141</sup></a><i>Op. cit.</i>, p. 658.</p> + +<p><a id="FN-142" href="#FNA-142"><sup>142</sup></a>Bancroft, H. H., <i>Native Races</i>, Vol. II. p. 499.</p> + +<p><a id="FN-143" href="#FNA-143"><sup>143</sup></a><i>Tr. Ethnological Soc. of London</i>, Vol. IV. p. 92.</p> + +<p><a id="FN-144" href="#FNA-144"><sup>144</sup></a>Any Hebrew lexicon.</p> + +<p><a id="FN-145" href="#FNA-145"><sup>145</sup></a>Schröder, P., <i>Die Phönizische Sprache, </i>p. 184 <i>et seq.</i></p> + +<p><a id="FN-146" href="#FNA-146"><sup>146</sup></a>Müller, <i>Sprachwissenschaft</i>, II. ii. p. 147.</p> + +<p><a id="FN-147" href="#FNA-147"><sup>147</sup></a><i>On Numerals in Am. Indian Languages.</i></p> + +<p><a id="FN-148" href="#FNA-148"><sup>148</sup></a>Ellis, A. B., <i>Ewe Speaking Peoples</i>, etc., p. 253. The meanings +here given are partly conjectural.</p> + +<p><a id="FN-149" href="#FNA-149"><sup>149</sup></a>Pott, <i>Zählmethode</i>, p. 29.</p> + +<p><a id="FN-150" href="#FNA-150"><sup>150</sup></a>Schoolcraft, <i>op. cit.</i>, Vol. IV. p. 429.</p> + +<p><a id="FN-151" href="#FNA-151"><sup>151</sup></a>Trumbull, <i>op. cit.</i></p> + +<p><a id="FN-152" href="#FNA-152"><sup>152</sup></a>Chamberlain, A. F., <i>Lang, of the Mississaga Indians</i>, Vocab.</p> + +<p><a id="FN-153" href="#FNA-153"><sup>153</sup></a>Crawfurd, <i>Hist. Ind. Archipelago</i>, 1, p. 258.</p> + +<p><a id="FN-154" href="#FNA-154"><sup>154</sup></a>Hale, H., <i>Eth. and Philol.</i>, Vol. VII.; Wilkes, <i>Expl. Expedition</i>, Phil. +1846, p. 172.</p> + +<p><a id="FN-155" href="#FNA-155"><sup>155</sup></a>Crawfurd, <i>op. cit.</i>, 1, p. 258.</p> + +<p><a id="FN-156" href="#FNA-156"><sup>156</sup></a><i>Op. cit.</i>, <i>loc. cit.</i></p> + +<p><a id="FN-157" href="#FNA-157"><sup>157</sup></a>Bancroft, H. H., <i>Native Races</i>, Vol. II. p. 498.</p> + +<p><a id="FN-158" href="#FNA-158"><sup>158</sup></a>Vignoli, T., <i>Myth and Science</i>, p. 203.</p> + +<p><a id="FN-159" href="#FNA-159"><sup>159</sup></a>Codrington, R. H., <i>The Melanesian Languages</i>, p. 249.</p> + +<p><a id="FN-160" href="#FNA-160"><sup>160</sup></a><i>Op. cit.</i>, <i>loc. cit.</i></p> + +<p><a id="FN-161" href="#FNA-161"><sup>161</sup></a>Codrington, R. H., <i>The Melanesian Languages</i>, p. 249.</p> + +<p><a id="FN-162" href="#FNA-162"><sup>162</sup></a>Wickersham, J., “Japanese Art on Puget Sound,” <i>Am. Antiq.</i>, 1894, +p. 79.</p> + +<p><a id="FN-163" href="#FNA-163"><sup>163</sup></a>Codrington, R. H., <i>op. cit.</i>, p. 250.</p> + +<p><a id="FN-164" href="#FNA-164"><sup>164</sup></a>Tylor, <i>Primitive Culture</i>, Vol. I. p. 252.</p> + +<p><a id="FN-165" href="#FNA-165"><sup>165</sup></a>Compare a similar table by Chase, <i>Proc. Amer. Philos. Soc.</i>, +1865, p. 23.</p> + +<p><a id="FN-166" href="#FNA-166"><sup>166</sup></a><i>Leibnitzii Opera</i>, III. p. 346.</p> + +<p><a id="FN-167" href="#FNA-167"><sup>167</sup></a>Pruner-Bey, <i>Bulletin de la Soc. d'Anthr. de Paris</i>, 1860, p. 486.</p> + +<p><a id="FN-168" href="#FNA-168"><sup>168</sup></a>Curr, E. M., <i>The Australian Race</i>, Vol. I. p. 32.</p> + +<p><a id="FN-169" href="#FNA-169"><sup>169</sup></a>Haddon, A. C., “Western Tribes of the Torres Straits,” <i>Journ. Anthr. +Inst.</i>, 1889, p. 303.</p> + +<p><a id="FN-170" href="#FNA-170"><sup>170</sup></a>Taplin, Rev. G., “Notes on a Table of Australian Languages,” <i>Journ. +Anthr. Inst.,</i> 1872, p. 88. The first nine scales are taken from this source.</p> + +<p><a id="FN-171" href="#FNA-171"><sup>171</sup></a>Latham, R. G., <i>Comparative Philology</i>, p. 352.</p> + +<p><a id="FN-172" href="#FNA-172"><sup>172</sup></a>It will be observed that this list differs slightly from that given in +Chapter II.</p> + +<p><a id="FN-173" href="#FNA-173"><sup>173</sup></a>Curr, E. M., <i>The Australian Race</i>, Vol. III. p. 684.</p> + +<p><a id="FN-174" href="#FNA-174"><sup>174</sup></a>Bonwick, <i>Tasmania</i>, p. 143.</p> + +<p><a id="FN-175" href="#FNA-175"><sup>175</sup></a>Lang, J. D., <i>Queensland</i>, p. 435.</p> + +<p><a id="FN-176" href="#FNA-176"><sup>176</sup></a>Bonwick, <i>Tasmania</i>, p. 143.</p> + +<p><a id="FN-177" href="#FNA-177"><sup>177</sup></a>Müller, <i>Sprachwissenschaft</i>, II. +i. p. 58.</p> + +<p><a id="FN-178" href="#FNA-178"><sup>178</sup></a><i>Op. cit.</i>, II. i. p. 70.</p> + +<p><a id="FN-179" href="#FNA-179"><sup>179</sup></a><i>Op. cit.</i>, II. i. p. 23.</p> + +<p><a id="FN-180" href="#FNA-180"><sup>180</sup></a>Barlow, H., “Aboriginal Dialects +of Queensland,” <i>Journ. Anth. +Inst.</i>, 1873, p. 171.</p> + +<p><a id="FN-181" href="#FNA-181"><sup>181</sup></a>Curr, E. M., <i>The Australian Race</i>, Vol. II. p. 26.</p> + +<p><a id="FN-182" href="#FNA-182"><sup>182</sup></a><i>Op. cit.</i>, Vol. II. p. 208.</p> + +<p><a id="FN-183" href="#FNA-183"><sup>183</sup></a><i>Op. cit.</i>, Vol. II. p. 278.</p> + +<p><a id="FN-184" href="#FNA-184"><sup>184</sup></a><i>Op. cit.</i>, Vol. II. p. 288.</p> + +<p><a id="FN-185" href="#FNA-185"><sup>185</sup></a><i>Op. cit.</i>, Vol. I. p. 258.</p> + +<p><a id="FN-186" href="#FNA-186"><sup>186</sup></a><i>Op. cit.</i>, Vol. I. p. 316.</p> + +<p><a id="FN-187" href="#FNA-187"><sup>187</sup></a><i>Op. cit.</i>, Vol. III. p. 32. The next ten lists are taken from the same +volume, pp. 282, 288, 340, 376, 432, 506, 530, 558, 560, 588, respectively.</p> + +<p><a id="FN-188" href="#FNA-188"><sup>188</sup></a>Brinton, <i>The American Race</i>, p. 351.</p> + +<p><a id="FN-189" href="#FNA-189"><sup>189</sup></a>Martius, <i>Glossaria Ling. Brazil.</i>, p. 307.</p> + +<p><a id="FN-190" href="#FNA-190"><sup>190</sup></a><i>Op. cit.</i>, p. 148.</p> + +<p><a id="FN-191" href="#FNA-191"><sup>191</sup></a>Müller, <i>Sprachwissenschaft</i>, II. i. p. 438.</p> + +<p><a id="FN-192" href="#FNA-192"><sup>192</sup></a>Peacock, “Arithmetic,” <i>Encyc. Metropolitana</i>, 1, p. 480.</p> + +<p><a id="FN-193" href="#FNA-193"><sup>193</sup></a>Brinton, <i>Studies in So. Am. Native Langs.</i>, p. 67.</p> + +<p><a id="FN-194" href="#FNA-194"><sup>194</sup></a><i>Op. cit.</i>, <i>loc. cit.</i></p> + +<p><a id="FN-195" href="#FNA-195"><sup>195</sup></a>Brinton, <i>Studies in So. Am. Native Langs.</i>, p. 67. The meanings of +the numerals are from Peacock, <i>Encyc. Metropolitana</i>, 1, p. 480.</p> + +<p><a id="FN-196" href="#FNA-196"><sup>196</sup></a>Mason, <i>Journ. As. Soc. of Bengal</i>, Vol. XXVI. p. 146.</p> + +<p><a id="FN-197" href="#FNA-197"><sup>197</sup></a>Curr, E. M., <i>The Australian Race</i>, Vol. III. p. 108.</p> + +<p><a id="FN-198" href="#FNA-198"><sup>198</sup></a>Bancroft, H. H., <i>Native Races</i>, Vol. I. p. 274.</p> + +<p><a id="FN-199" href="#FNA-199"><sup>199</sup></a>Clarke, Hyde, <i>Journ. Anthr. Inst.</i>, 1872, p. clvii. In the article +from which this is quoted, no evidence is given to substantiate the assertion +made. It is to be received with great caution.</p> + +<p><a id="FN-200" href="#FNA-200"><sup>200</sup></a>Hale, H., <i>Wilkes Exploring Expedition</i>, Vol. VII. p. 172.</p> + +<p><a id="FN-201" href="#FNA-201"><sup>201</sup></a><i>Op. cit.</i>, p. 248.</p> + +<p><a id="FN-202" href="#FNA-202"><sup>202</sup></a>Hale, <i>Ethnography and Philology, </i>p. 247.</p> + +<p><a id="FN-203" href="#FNA-203"><sup>203</sup></a><i>Loc. cit.</i></p> + +<p><a id="FN-204" href="#FNA-204"><sup>204</sup></a>Ellis, <i>Polynesian Researches</i>, Vol. IV. p. 341.</p> + +<p><a id="FN-205" href="#FNA-205"><sup>205</sup></a>Gill, W. W., <i>Myths and Songs of the South Pacific</i>, p. 325.</p> + +<p><a id="FN-206" href="#FNA-206"><sup>206</sup></a>Peacock, “Arithmetic,” <i>Encyc. Metropolitana</i>, 1, p. 479.</p> + +<p><a id="FN-207" href="#FNA-207"><sup>207</sup></a>Peacock, <i>Encyc. Metropolitana</i>, 1, p. 480.</p> + +<p><a id="FN-208" href="#FNA-208"><sup>208</sup></a><i>Sprachverschiedenheit</i>, p. 30.</p> + +<p><a id="FN-209" href="#FNA-209"><sup>209</sup></a>Crawfurd, <i>History of the Indian Archipelago</i>, Vol. I. p. 256.</p> + +<p><a id="FN-210" href="#FNA-210"><sup>210</sup></a>Pott, <i>Zählmethode</i>, p. 39.</p> + +<p><a id="FN-211" href="#FNA-211"><sup>211</sup></a><i>Op. cit.</i>, p. 41.</p> + +<p><a id="FN-212" href="#FNA-212"><sup>212</sup></a>Müller, <i>Sprachwissenschaft</i>, II. i. p. 317. See also Chap. III., <i>supra</i>.</p> + +<p><a id="FN-213" href="#FNA-213"><sup>213</sup></a>Long, S. H., <i>Expedition</i>, Vol. II. p. lxxviii.</p> + +<p><a id="FN-214" href="#FNA-214"><sup>214</sup></a>Martius, <i>Glossaria Ling. Brasil.</i>, p. 246.</p> + +<p><a id="FN-215" href="#FNA-215"><sup>215</sup></a>Hale, <i>Ethnography and Philology</i>, p. 434.</p> + +<p><a id="FN-216" href="#FNA-216"><sup>216</sup></a>Müller, <i>Sprachwissenschaft</i>, II. ii. p. 82.</p> + +<p><a id="FN-217" href="#FNA-217"><sup>217</sup></a>The information upon which the above statements are based was +obtained from Mr. W. L. Williams, of Gisborne, N.Z.</p> + +<p><a id="FN-218" href="#FNA-218"><sup>218</sup></a><i>Primitive Culture</i>, Vol. I. p. 268.</p> + +<p><a id="FN-219" href="#FNA-219"><sup>219</sup></a>Ralph, Julian, <i>Harper's Monthly</i>, Vol. 86, p. 184.</p> + +<p><a id="FN-220" href="#FNA-220"><sup>220</sup></a>Lappenberg, J. M., <i>History of Eng. under the Anglo-Saxon Kings</i>, +Vol. I. p. 82.</p> + +<p><a id="FN-221" href="#FNA-221"><sup>221</sup></a>The compilation of this table was suggested by a comparison found +in the <i>Bulletin Soc. Anth. de Paris</i>, 1886, p. 90.</p> + +<p><a id="FN-222" href="#FNA-222"><sup>222</sup></a>Hale, <i>Ethnography and Philology</i>, p. 126.</p> + +<p><a id="FN-223" href="#FNA-223"><sup>223</sup></a>Müller, <i>Sprachwissenschaft</i>, II. ii. p. 183.</p> + +<p><a id="FN-224" href="#FNA-224"><sup>224</sup></a>Bachofen, J. J., <i>Antiquarische Briefe</i>, Vol. I. pp. 101–115, and Vol. +II. pp. 1–90.</p> + +<p><a id="FN-225" href="#FNA-225"><sup>225</sup></a>An extended table of this kind may be found in the last part of +Nystrom's <i>Mechanics</i>.</p> + +<p><a id="FN-226" href="#FNA-226"><sup>226</sup></a>Schubert, H., quoting Robert Flegel, in Neumayer's <i>Anleitung zu +Wissenschaftlichen Beobachtung auf Reisen</i>, Vol. II. p. 290.</p> + +<p><a id="FN-227" href="#FNA-227"><sup>227</sup></a>These numerals, and those in all the sets immediately following, +except those for which the authority is given, are to be found in Chapter +III.</p> + +<p><a id="FN-228" href="#FNA-228"><sup>228</sup></a>Codrington, <i>The Melanesian Languages</i>, p. 222.</p> + +<p><a id="FN-229" href="#FNA-229"><sup>229</sup></a>Müller, <i>Sprachwissenschaft</i>, II. ii. p. 83.</p> + +<p><a id="FN-230" href="#FNA-230"><sup>230</sup></a><i>Op. cit.</i>, I. ii. p. 55. The next two are the same, p. 83 and p. 210. +The meaning given for the Bari <i>puök</i> is wholly conjectural.</p> + +<p><a id="FN-231" href="#FNA-231"><sup>231</sup></a>Gallatin, “Semi-civilized Nations,” <i>Tr. Am. Eth. Soc.</i>, Vol. I. p. 114.</p> + +<p><a id="FN-232" href="#FNA-232"><sup>232</sup></a>Müller, <i>Sprachwissenschaft</i>, II. ii. p. 80. Erromango, the same.</p> + +<p><a id="FN-233" href="#FNA-233"><sup>233</sup></a>Boas, Fr., <i>Proc. Brit. Ass'n. Adv. Science</i>, 1889, p. 857.</p> + +<p><a id="FN-234" href="#FNA-234"><sup>234</sup></a>Hankel, H., <i>Geschichte der Mathematik</i>, p. 20.</p> + +<p><a id="FN-235" href="#FNA-235"><sup>235</sup></a>Murdoch, J., “Eskimos of Point Barrow,” <i>Am. Anthr.</i>, 1890, p. 40.</p> + +<p><a id="FN-236" href="#FNA-236"><sup>236</sup></a>Martius, <i>Glos. Ling. Brasil.</i>, p. 360.</p> + +<p><a id="FN-237" href="#FNA-237"><sup>237</sup></a>Du Graty, A. M., <i>La République du Paraguay</i>, p. 217.</p> + +<p><a id="FN-238" href="#FNA-238"><sup>238</sup></a>Codrington, <i>The Melanesian Languages</i>, p. 221.</p> + +<p><a id="FN-239" href="#FNA-239"><sup>239</sup></a>Müller, <i>Sprachwissenschaft</i>, II. i. p. 363.</p> + +<p><a id="FN-240" href="#FNA-240"><sup>240</sup></a>Spurrell, W., <i>Welsh Grammar</i>, p. 59.</p> + +<p><a id="FN-241" href="#FNA-241"><sup>241</sup></a>Olmos, André de, <i>Grammaire Nahuatl ou Mexicaine</i>, p. 191.</p> + +<p><a id="FN-242" href="#FNA-242"><sup>242</sup></a>Moncelon, <i>Bull. Soc. d'Anthr. de Paris</i>, 1885, p. 354. This is a +purely digital scale, but unfortunately M. Moncelon does not give the +meanings of any of the numerals except the last.</p> + +<p><a id="FN-243" href="#FNA-243"><sup>243</sup></a>Ellis, <i>Peruvia Scythia</i>, p. 37. Part of these numerals are from Martius, +<i>Glos. Brasil.</i>, p. 210.</p> + +<p><a id="FN-244" href="#FNA-244"><sup>244</sup></a>Codrington, <i>The Melanesian Languages</i>, p. 236.</p> + +<p><a id="FN-245" href="#FNA-245"><sup>245</sup></a>Schweinfurth, G., <i>Linguistische Ergebnisse einer Reise nach Centralafrika</i>, +p. 25.</p> + +<p><a id="FN-246" href="#FNA-246"><sup>246</sup></a>Park, M., <i>Travels in the Interior Districts of Africa</i>, p. 8.</p> + +<p><a id="FN-247" href="#FNA-247"><sup>247</sup></a>Pott, <i>Zählmethode</i>, p. 37.</p> + +<p><a id="FN-248" href="#FNA-248"><sup>248</sup></a><i>Op. cit.</i>, p. 39.</p> + +<p><a id="FN-249" href="#FNA-249"><sup>249</sup></a>Müller, <i>Sprachwissenschaft</i>, IV. i. p. 101. The Kru scale, kindred +with the Basa, is from the same page.</p> + +<p><a id="FN-250" href="#FNA-250"><sup>250</sup></a>Park, in Pinkerton's <i>Voyages and Travels</i>, Vol. XVI. p. 902.</p> + +<p><a id="FN-251" href="#FNA-251"><sup>251</sup></a>Park, <i>Travels</i>, Vol. I. p. 16.</p> + +<p><a id="FN-252" href="#FNA-252"><sup>252</sup></a>Schweinfurth, G., <i>Linguistische Ergebnisse einer Reise nach Centralafrika</i>, +p. 78.</p> + +<p><a id="FN-253" href="#FNA-253"><sup>253</sup></a>Park, <i>Travels</i>, Vol. I. p. 58.</p> + +<p><a id="FN-254" href="#FNA-254"><sup>254</sup></a>Goedel, “Ethnol. des Soussous,” <i>Bull. Soc. Anth. Paris</i>, 1892, p. 185.</p> + +<p><a id="FN-255" href="#FNA-255"><sup>255</sup></a>Müller, <i>Sprachwissenschaft</i>, I. ii. p. 114. The Temne scale is from +the same page. These two languages are closely related.</p> + +<p><a id="FN-256" href="#FNA-256"><sup>256</sup></a><i>Op. cit.</i>, I. ii. p. 155.</p> + +<p><a id="FN-257" href="#FNA-257"><sup>257</sup></a><i>Op. cit.</i>, I. ii. p. 55.</p> + +<p><a id="FN-258" href="#FNA-258"><sup>258</sup></a>Long, C. C., <i>Central Africa</i>, p. 330.</p> + +<p><a id="FN-259" href="#FNA-259"><sup>259</sup></a>Müller, <i>Sprachwissenschaft</i>, IV. i. p. 105.</p> + +<p><a id="FN-260" href="#FNA-260"><sup>260</sup></a>Pott, <i>Zählmethode</i>, p. 41.</p> + +<p><a id="FN-261" href="#FNA-261"><sup>261</sup></a>Müller, <i>op. cit.</i>, I. ii. p. 140.</p> + +<p><a id="FN-262" href="#FNA-262"><sup>262</sup></a>Müller, <i>Sprachwissenschaft</i>, IV. i. p. 81.</p> + +<p><a id="FN-263" href="#FNA-263"><sup>263</sup></a>Pott, <i>Zählmethode</i>, p. 41.</p> + +<p><a id="FN-264" href="#FNA-264"><sup>264</sup></a>Müller, <i>op. cit.</i>, I. ii., p. 210.</p> + +<p><a id="FN-265" href="#FNA-265"><sup>265</sup></a>Pott, <i>Zählmethode</i>, p. 42.</p> + +<p><a id="FN-266" href="#FNA-266"><sup>266</sup></a>Schweinfurth, <i>Linguistische Ergebnisse</i>, p. 59.</p> + +<p><a id="FN-267" href="#FNA-267"><sup>267</sup></a>Müller, <i>Sprachwissenschaft</i>, I. ii. p. 261. The “ten” is not given.</p> + +<p><a id="FN-268" href="#FNA-268"><sup>268</sup></a>Stanley, <i>Through the Dark Continent</i>, Vol. II. p. 490. Ki-Nyassa, +the same page.</p> + +<p><a id="FN-269" href="#FNA-269"><sup>269</sup></a>Müller, <i>op. cit.</i>, I. ii. p. 261.</p> + +<p><a id="FN-270" href="#FNA-270"><sup>270</sup></a>Du Chaillu, <i>Adventures in Equatorial Africa</i>, p. 534.</p> + +<p><a id="FN-271" href="#FNA-271"><sup>271</sup></a>Müller, <i>Sprachwissenschaft</i>, III. i. p. 65.</p> + +<p><a id="FN-272" href="#FNA-272"><sup>272</sup></a>Du Chaillu, <i>Adventures in Equatorial Africa</i>, p. 533.</p> + +<p><a id="FN-273" href="#FNA-273"><sup>273</sup></a>Müller, <i>op. cit.</i>, III. ii. p. 77.</p> + +<p><a id="FN-274" href="#FNA-274"><sup>274</sup></a>Balbi, A., <i>L'Atlas Eth.</i>, Vol. I. p. 226. In Balbi's text 7 and 8 are +ansposed. <i>Taru</i> for 5 is probably a misprint for <i>tana</i>.</p> + +<p><a id="FN-275" href="#FNA-275"><sup>275</sup></a>Du Chaillu, <i>op. cit.</i>, p. 533. The next scale is <i>op. cit.</i>, p. 534.</p> + +<p><a id="FN-276" href="#FNA-276"><sup>276</sup></a>Beauregard, O., <i>Bull. Soc. Anth. de Paris</i>, 1886, p. 526.</p> + +<p><a id="FN-277" href="#FNA-277"><sup>277</sup></a>Pott, <i>Zählmethode</i>, p. 46.</p> + +<p><a id="FN-278" href="#FNA-278"><sup>278</sup></a><i>Op. cit.</i>, p. 48.</p> + +<p><a id="FN-279" href="#FNA-279"><sup>279</sup></a>Turner, <i>Nineteen Years in Polynesia</i>, p. 536.</p> + +<p><a id="FN-280" href="#FNA-280"><sup>280</sup></a>Erskine, J. E., <i>Islands of the Western Pacific</i>, p. 341.</p> + +<p><a id="FN-281" href="#FNA-281"><sup>281</sup></a><i>Op. cit.</i>, p. 400.</p> + +<p><a id="FN-282" href="#FNA-282"><sup>282</sup></a>Codrington, <i>Melanesian Languages</i>, pp. 235, 236.</p> + +<p><a id="FN-283" href="#FNA-283"><sup>283</sup></a>Peacock, <i>Encyc. Met.</i>, Vol. 1. p. 385. Peacock does not specify the +dialect.</p> + +<p><a id="FN-284" href="#FNA-284"><sup>284</sup></a>Erskine, <i>Islands of the Western Pacific</i>, p. 360.</p> + +<p><a id="FN-285" href="#FNA-285"><sup>285</sup></a>Turner, G., <i>Samoa a Hundred Years Ago</i>, p. 373. The next three +scales are from the same page of this work.</p> + +<p><a id="FN-286" href="#FNA-286"><sup>286</sup></a>Codrington, <i>Melanesian Languages</i>, p. 235. The next four scales +are from the same page. Perhaps the meanings of the words for 6 to 9 +are more properly “more 1,” “more 2,” etc. Codrington merely +indicates their significations in a general way.</p> + +<p><a id="FN-287" href="#FNA-287"><sup>287</sup></a>Hale, <i>Ethnography and Philology</i>, p. 429. The meanings of 6 to 9 +in this and the preceding are my conjectures.</p> + +<p><a id="FN-288" href="#FNA-288"><sup>288</sup></a>Müller, <i>Sprachwissenschaft</i>, IV. i. p. 124.</p> + +<p><a id="FN-289" href="#FNA-289"><sup>289</sup></a>Aymonier, E., <i>Dictionnaire Francaise-Cambodgien</i>.</p> + +<p><a id="FN-290" href="#FNA-290"><sup>290</sup></a>Müller, <i>Op. cit.</i>, II. i. p. 139.</p> + +<p><a id="FN-291" href="#FNA-291"><sup>291</sup></a>Müller, <i>Sprachwissenschaft</i>, II. i. p. 123.</p> + +<p><a id="FN-292" href="#FNA-292"><sup>292</sup></a>Wells, E. R., Jr., and John W. Kelly, Bureau of Ed., Circ. of Inf., +No. 2, 1890.</p> + +<p><a id="FN-293" href="#FNA-293"><sup>293</sup></a>Pott, <i>Zählmethode</i>, p. 57.</p> + +<p><a id="FN-294" href="#FNA-294"><sup>294</sup></a>Müller, <i>Op. cit.</i>, II. i. p. 161.</p> + +<p><a id="FN-295" href="#FNA-295"><sup>295</sup></a>Petitot, <i>Vocabulaire Française Esquimau</i>, p. lv.</p> + +<p><a id="FN-296" href="#FNA-296"><sup>296</sup></a>Müller, <i>Sprachwissenschaft</i>, II. i. p. 253.</p> + +<p><a id="FN-297" href="#FNA-297"><sup>297</sup></a>Müller, <i>Op. cit.</i>, II. <del title="should be lower case roman numeral">I.</del><ins title="changed case of roman numeral, cf. other references to Müller">i.</ins> p. 179, and Kleinschmidt, <i title="sic!">Grönlandisches +Grammatik</i>.</p> + +<p><a id="FN-298" href="#FNA-298"><sup>298</sup></a>Adam, L., <i>Congres Int. des Am.</i>, 1877, p. 244 (see <a href="#page-162">p. 162 <i>infra</i></a>).</p> + +<p><a id="FN-299" href="#FNA-299"><sup>299</sup></a>Gallatin, “Synopsis of Indian Tribes,” <i>Trans. Am. Antq. Soc.</i>, 1836, p. +358. The next fourteen lists are, with the exception of the Micmac, from +the same collection. The meanings are largely from Trumbull, <i>op. cit.</i></p> + +<p><a id="FN-300" href="#FNA-300"><sup>300</sup></a>Schoolcraft, <i>Archives of Aboriginal Knowledge</i>, Vol. II. p. 211.</p> + +<p><a id="FN-301" href="#FNA-301"><sup>301</sup></a>Schoolcraft, <i>Archives of Aboriginal Knowledge</i>, Vol. V. p. 587.</p> + +<p><a id="FN-302" href="#FNA-302"><sup>302</sup></a>In the Dakota dialects 10 is expressed, as here, by a word signifying +that the fingers, which have been bent down in counting, are now straightened +out.</p> + +<p><a id="FN-303" href="#FNA-303"><sup>303</sup></a>Boas, <i>Fifth Report B. A. A. S.</i>, 1889. Reprint, p. 61.</p> + +<p><a id="FN-304" href="#FNA-304"><sup>304</sup></a>Boas, <i>Sixth Report B. A. A. S.</i>, 1890. Reprint, p. 117. Dr. Boas +does not give the meanings assigned to 7 and 8, but merely states +that they are derived from 2 and 3.</p> + +<p><a id="FN-305" href="#FNA-305"><sup>305</sup></a><i>Op. cit.</i>, p. 117. The derivations for 6 and 7 are obvious, but the +meanings are conjectural.</p> + +<p><a id="FN-306" href="#FNA-306"><sup>306</sup></a>Boas, <i>Sixth Report B. A. A. S.</i>, 1889. Reprint, pp. 158, 160. The +meanings assigned to the Tsimshian 8 and to Bilqula 6 to 8 are conjectural.</p> + +<p><a id="FN-307" href="#FNA-307"><sup>307</sup></a>Hale, <i>Ethnography and Philology</i>, p. 619.</p> + +<p><a id="FN-308" href="#FNA-308"><sup>308</sup></a><i>Op. cit.</i>, <i>loc. cit.</i></p> + +<p><a id="FN-309" href="#FNA-309"><sup>309</sup></a>Hale, <i>Ethnography and Philology</i>, p. 619.</p> + +<p><a id="FN-310" href="#FNA-310"><sup>310</sup></a>Müller, <i>Sprachwissenschaft</i>, II. i. p. 436.</p> + +<p><a id="FN-311" href="#FNA-311"><sup>311</sup></a><i>Op. cit.</i>, IV. i. p. 167.</p> + +<p><a id="FN-312" href="#FNA-312"><sup>312</sup></a><i>Op. cit.</i>, II. i. p. 282.</p> + +<p><a id="FN-313" href="#FNA-313"><sup>313</sup></a><i>Op. cit.</i>, II. i. p. 287. The meanings given for the words for 7, 8, 9 are +conjectures of my own.</p> + +<p><a id="FN-314" href="#FNA-314"><sup>314</sup></a>Müller, <i>Sprachwissenschaft</i>, II. i. p. 297.</p> + +<p><a id="FN-315" href="#FNA-315"><sup>315</sup></a>Pott, <i>Zählmethode</i>, p. 90.</p> + +<p><a id="FN-316" href="#FNA-316"><sup>316</sup></a>Müller, <i>op. cit.</i>, II. i. p. 379.</p> + +<p><a id="FN-317" href="#FNA-317"><sup>317</sup></a>Gallatin, “Semi-Civilized Nations of Mexico and Central America,” +<i>Tr. Am. Ethn. Soc.</i>, Vol. I. p. 114.</p> + +<p><a id="FN-318" href="#FNA-318"><sup>318</sup></a>Adam, Lucien, <i>Congres Internationale des Americanistes</i>, 1877, Vol. +II. p. 244.</p> + +<p><a id="FN-319" href="#FNA-319"><sup>319</sup></a>Müller, <i>Sprachwissenschaft</i>, II. i. p. 395. I can only guess at the +meanings of 6 to 9. They are obviously circumlocutions for 5-1, 5-2, etc.</p> + +<p><a id="FN-320" href="#FNA-320"><sup>320</sup></a><i>Op. cit.</i>, p. 438. Müller has transposed these two scales. See Brinton's +<i>Am. Race</i>, p. 358.</p> + +<p><a id="FN-321" href="#FNA-321"><sup>321</sup></a>Marcoy, P., <i>Tour du Monde</i>, 1866, 2ème sem. p. 148.</p> + +<p><a id="FN-322" href="#FNA-322"><sup>322</sup></a><i>Op. cit.</i>, p. 132. The meanings are my own conjectures.</p> + +<p><a id="FN-323" href="#FNA-323"><sup>323</sup></a>An elaborate argument in support of this theory is to be found +in Hervas' celebrated work, <i>Arithmetica di quasi tutte le nazioni conosciute</i>.</p> + +<p><a id="FN-324" href="#FNA-324"><sup>324</sup></a>See especially the lists of Hale, Gallatin, Trumbull, and Boas, to +which references have been given above.</p> + +<p><a id="FN-325" href="#FNA-325"><sup>325</sup></a>Thiel, B. A., “Vocab. der Indianier in Costa Rica,” <i>Archiv für +Anth.</i>, xvi. p. 620.</p> + +<p><a id="FN-326" href="#FNA-326"><sup>326</sup></a>These three examples are from A. R. Wallace's <i>Narrative of Travels +on the Amazon and Rio Negro</i>, vocab. Similar illustrations may be +found in Martius' <i>Glos. Brasil</i>.</p> + +<p><a id="FN-327" href="#FNA-327"><sup>327</sup></a>Martius, <i>Glos. Brasil.</i>, p. 176.</p> + +<p><a id="FN-328" href="#FNA-328"><sup>328</sup></a>Adam, L., <i>Congres International des Americanistes</i>, 1877, Vol. II. +p. 244. Given also <i>supra</i>, <a href="#page-53">p. 53</a>.</p> + +<p><a id="FN-329" href="#FNA-329"><sup>329</sup></a>O'Donovan, <i>Irish Grammar</i>, p. 123.</p> + +<p><a id="FN-330" href="#FNA-330"><sup>330</sup></a>Armstrong, R. A., <i>Gaelic Dict.</i>, p. xxi.</p> + +<p><a id="FN-331" href="#FNA-331"><sup>331</sup></a>Spurrell, <i>Welsh Dictionary</i>.</p> + +<p><a id="FN-332" href="#FNA-332"><sup>332</sup></a>Kelly, <i>Triglot Dict.</i>, pub. by the Manx Society.</p> + +<p><a id="FN-333" href="#FNA-333"><sup>333</sup></a>Guillome, J., <i>Grammaire Française-Bretonne</i>, p. 27.</p> + +<p><a id="FN-334" href="#FNA-334"><sup>334</sup></a>Gröber, G., <i>Grundriss der Romanischen Philologie</i>, Bd. I. p. 309.</p> + +<p><a id="FN-335" href="#FNA-335"><sup>335</sup></a>Pott, <i>Zählmethode</i>, p. 88.</p> + +<p><a id="FN-336" href="#FNA-336"><sup>336</sup></a>Van Eys, <i>Basque Grammar</i>, p. 27.</p> + +<p><a id="FN-337" href="#FNA-337"><sup>337</sup></a>Pott, <i>Zählmethode</i>, p. 101.</p> + +<p><a id="FN-338" href="#FNA-338"><sup>338</sup></a><i>Op. cit.</i>, p. 78.</p> + +<p><a id="FN-339" href="#FNA-339"><sup>339</sup></a>Müller, <i>Sprachwissenschaft</i>, I. ii. p. 124.</p> + +<p><a id="FN-340" href="#FNA-340"><sup>340</sup></a><i>Op. cit.</i>, p. 155.</p> + +<p><a id="FN-341" href="#FNA-341"><sup>341</sup></a><i>Op. cit.</i>, p. 140.</p> + +<p><a id="FN-342" href="#FNA-342"><sup>342</sup></a><i>Op. cit.</i>, <i>loc. cit.</i></p> + +<p><a id="FN-343" href="#FNA-343"><sup>343</sup></a>Schweinfurth, <i>Reise nach Centralafrika</i>, p. 25.</p> + +<p><a id="FN-344" href="#FNA-344"><sup>344</sup></a>Müller, <i>Sprachwissenschaft</i>, IV. i. p. 83.</p> + +<p><a id="FN-345" href="#FNA-345"><sup>345</sup></a><i>Op. cit.</i>, IV. i. p. 81.</p> + +<p><a id="FN-346" href="#FNA-346"><sup>346</sup></a><i>Op. cit.</i>, I. ii. p. 166.</p> + +<p><a id="FN-347" href="#FNA-347"><sup>347</sup></a>Long, C. C., <i>Central Africa</i>, p. 330.</p> + +<p><a id="FN-348" href="#FNA-348"><sup>348</sup></a>Peacock, <i>Encyc. Met.</i>, Vol. I. p. 388.</p> + +<p><a id="FN-349" href="#FNA-349"><sup>349</sup></a>Müller, <i>Sprachwissenschaft</i>, III. ii. p. 64. The next seven scales +are from <i>op. cit.</i>, pp. 80, 137, 155, 182, 213.</p> + +<p><a id="FN-350" href="#FNA-350"><sup>350</sup></a>Pott, <i>Zählmethode</i>, p. 83.</p> + +<p><a id="FN-351" href="#FNA-351"><sup>351</sup></a><i>Op. cit.</i>, p. 83,—Akari, p. 84; Circassia, p. 85.</p> + +<p><a id="FN-352" href="#FNA-352"><sup>352</sup></a>Müller, <i>Sprachwissenschaft</i>, II. i. p. 140.</p> + +<p><a id="FN-353" href="#FNA-353"><sup>353</sup></a>Pott, <i>Zählmethode</i>, p. 87.</p> + +<p><a id="FN-354" href="#FNA-354"><sup>354</sup></a>Müller, <i>Sprachwissenschaft</i>, II. ii. p. 346.</p> + +<p><a id="FN-355" href="#FNA-355"><sup>355</sup></a><i>Op. cit.</i>, III. i. p. 130.</p> + +<p><a id="FN-356" href="#FNA-356"><sup>356</sup></a>Man, E. H., “Brief Account of the Nicobar Islands,” <i>Journ. Anthr. +Inst.</i>, 1885, p. 435.</p> + +<p><a id="FN-357" href="#FNA-357"><sup>357</sup></a>Wells, E. R., Jr., and Kelly, J. W., “Eng. Esk. and Esk. Eng. +Vocab.,” Bureau of Education Circular of Information, No. 2, 1890, p. 65.</p> + +<p><a id="FN-358" href="#FNA-358"><sup>358</sup></a>Petitot, E., <i>Vocabulaire Française Esquimau</i>, p. lv.</p> + +<p><a id="FN-359" href="#FNA-359"><sup>359</sup></a>Boas, Fr., <i>Proc. Brit. Ass. Adv. Sci.</i>, 1889, p. 857.</p> + +<p><a id="FN-360" href="#FNA-360"><sup>360</sup></a>Boas, <i>Sixth Report on the Northwestern Tribes of Canada</i>, p. 117.</p> + +<p><a id="FN-361" href="#FNA-361"><sup>361</sup></a>Boas, Fr., <i>Fifth Report on the Northwestern Tribes of Canada</i>, p. 85.</p> + +<p><a id="FN-362" href="#FNA-362"><sup>362</sup></a>Gallatin, <i>Semi-Civilized Nations</i>, p. 114. References for the next +two are the same.</p> + +<p><a id="FN-363" href="#FNA-363"><sup>363</sup></a>Bancroft, H. H., <i>Native Races of the Pacific States</i>, Vol. II. p. 763. +The meanings are from Brinton's <i>Maya Chronicles</i>, p. 38 <i>et seq.</i></p> + +<p><a id="FN-364" href="#FNA-364"><sup>364</sup></a>Brinton, <i>Maya Chronicles</i>, p. 44.</p> + +<p><a id="FN-365" href="#FNA-365"><sup>365</sup></a>Siméon Rémi, <i>Dictionnaire de la langue nahuatl</i>, p. xxxii.</p> + +<p><a id="FN-366" href="#FNA-366"><sup>366</sup></a>An error occurs on p. xxxiv of the work from which these numerals +are taken, which makes the number in question appear as 279,999,999 +instead of 1,279,999,999.</p> + +<p><a id="FN-367" href="#FNA-367"><sup>367</sup></a>Gallatin, “Semi-Civilized Nations of Mexico and Central America,” +<i>Tr. Am. Ethn. Soc.</i> Vol. I. p. 114.</p> + +<p><a id="FN-368" href="#FNA-368"><sup>368</sup></a>Pott, <i>Zählmethode</i>, p. 89. The Totonacos were the first race Cortez +encountered after landing in Mexico.</p> + +<p><a id="FN-369" href="#FNA-369"><sup>369</sup></a><i>Op. cit.</i>, p. 90. The Coras are of the Mexican state of Sonora.</p> + +<p><a id="FN-370" href="#FNA-370"><sup>370</sup></a>Gallatin, <i>Semi-Civilized Nations</i>, p. 114.</p> + +<p><a id="FN-371" href="#FNA-371"><sup>371</sup></a>Humboldt, <i>Recherches</i>, Vol. II. p. 112.</p> + +<p><a id="FN-372" href="#FNA-372"><sup>372</sup></a>Squier, <i>Nicaragua</i>, Vol. II. p. 326.</p> + +<p><a id="FN-373" href="#FNA-373"><sup>373</sup></a>Gallatin, <i>Semi-Civilized Nations</i>, p. 57.</p> + +</div> + + + + + + + +<pre> + + + + + +End of Project Gutenberg's The Number Concept, by Levi Leonard Conant + +*** END OF THIS PROJECT GUTENBERG EBOOK THE NUMBER CONCEPT *** + +***** This file should be named 16449-h.htm or 16449-h.zip ***** +This and all associated files of various formats will be found in: + https://www.gutenberg.org/1/6/4/4/16449/ + +Produced by Jonathan Ingram, Hagen von Eitzen and the +Online Distributed Proofreading Team at https://www.pgdp.net + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. Special rules, +set forth in the General Terms of Use part of this license, apply to +copying and distributing Project Gutenberg-tm electronic works to +protect the PROJECT GUTENBERG-tm concept and trademark. Project +Gutenberg is a registered trademark, and may not be used if you +charge for the eBooks, unless you receive specific permission. If you +do not charge anything for copies of this eBook, complying with the +rules is very easy. You may use this eBook for nearly any purpose +such as creation of derivative works, reports, performances and +research. They may be modified and printed and given away--you may do +practically ANYTHING with public domain eBooks. Redistribution is +subject to the trademark license, especially commercial +redistribution. + + + +*** START: FULL LICENSE *** + +THE FULL PROJECT GUTENBERG LICENSE +PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK + +To protect the Project Gutenberg-tm mission of promoting the free +distribution of electronic works, by using or distributing this work +(or any other work associated in any way with the phrase "Project +Gutenberg"), you agree to comply with all the terms of the Full Project +Gutenberg-tm License (available with this file or online at +https://gutenberg.org/license). + + +Section 1. General Terms of Use and Redistributing Project Gutenberg-tm +electronic works + +1.A. By reading or using any part of this Project Gutenberg-tm +electronic work, you indicate that you have read, understand, agree to +and accept all the terms of this license and intellectual property +(trademark/copyright) agreement. If you do not agree to abide by all +the terms of this agreement, you must cease using and return or destroy +all copies of Project Gutenberg-tm electronic works in your possession. +If you paid a fee for obtaining a copy of or access to a Project +Gutenberg-tm electronic work and you do not agree to be bound by the +terms of this agreement, you may obtain a refund from the person or +entity to whom you paid the fee as set forth in paragraph 1.E.8. + +1.B. "Project Gutenberg" is a registered trademark. It may only be +used on or associated in any way with an electronic work by people who +agree to be bound by the terms of this agreement. There are a few +things that you can do with most Project Gutenberg-tm electronic works +even without complying with the full terms of this agreement. See +paragraph 1.C below. There are a lot of things you can do with Project +Gutenberg-tm electronic works if you follow the terms of this agreement +and help preserve free future access to Project Gutenberg-tm electronic +works. See paragraph 1.E below. + +1.C. The Project Gutenberg Literary Archive Foundation ("the Foundation" +or PGLAF), owns a compilation copyright in the collection of Project +Gutenberg-tm electronic works. Nearly all the individual works in the +collection are in the public domain in the United States. If an +individual work is in the public domain in the United States and you are +located in the United States, we do not claim a right to prevent you from +copying, distributing, performing, displaying or creating derivative +works based on the work as long as all references to Project Gutenberg +are removed. Of course, we hope that you will support the Project +Gutenberg-tm mission of promoting free access to electronic works by +freely sharing Project Gutenberg-tm works in compliance with the terms of +this agreement for keeping the Project Gutenberg-tm name associated with +the work. You can easily comply with the terms of this agreement by +keeping this work in the same format with its attached full Project +Gutenberg-tm License when you share it without charge with others. + +1.D. The copyright laws of the place where you are located also govern +what you can do with this work. Copyright laws in most countries are in +a constant state of change. If you are outside the United States, check +the laws of your country in addition to the terms of this agreement +before downloading, copying, displaying, performing, distributing or +creating derivative works based on this work or any other Project +Gutenberg-tm work. The Foundation makes no representations concerning +the copyright status of any work in any country outside the United +States. + +1.E. Unless you have removed all references to Project Gutenberg: + +1.E.1. The following sentence, with active links to, or other immediate +access to, the full Project Gutenberg-tm License must appear prominently +whenever any copy of a Project Gutenberg-tm work (any work on which the +phrase "Project Gutenberg" appears, or with which the phrase "Project +Gutenberg" is associated) is accessed, displayed, performed, viewed, +copied or distributed: + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + +1.E.2. If an individual Project Gutenberg-tm electronic work is derived +from the public domain (does not contain a notice indicating that it is +posted with permission of the copyright holder), the work can be copied +and distributed to anyone in the United States without paying any fees +or charges. If you are redistributing or providing access to a work +with the phrase "Project Gutenberg" associated with or appearing on the +work, you must comply either with the requirements of paragraphs 1.E.1 +through 1.E.7 or obtain permission for the use of the work and the +Project Gutenberg-tm trademark as set forth in paragraphs 1.E.8 or +1.E.9. + +1.E.3. If an individual Project Gutenberg-tm electronic work is posted +with the permission of the copyright holder, your use and distribution +must comply with both paragraphs 1.E.1 through 1.E.7 and any additional +terms imposed by the copyright holder. Additional terms will be linked +to the Project Gutenberg-tm License for all works posted with the +permission of the copyright holder found at the beginning of this work. + +1.E.4. Do not unlink or detach or remove the full Project Gutenberg-tm +License terms from this work, or any files containing a part of this +work or any other work associated with Project Gutenberg-tm. + +1.E.5. Do not copy, display, perform, distribute or redistribute this +electronic work, or any part of this electronic work, without +prominently displaying the sentence set forth in paragraph 1.E.1 with +active links or immediate access to the full terms of the Project +Gutenberg-tm License. + +1.E.6. You may convert to and distribute this work in any binary, +compressed, marked up, nonproprietary or proprietary form, including any +word processing or hypertext form. However, if you provide access to or +distribute copies of a Project Gutenberg-tm work in a format other than +"Plain Vanilla ASCII" or other format used in the official version +posted on the official Project Gutenberg-tm web site (www.gutenberg.org), +you must, at no additional cost, fee or expense to the user, provide a +copy, a means of exporting a copy, or a means of obtaining a copy upon +request, of the work in its original "Plain Vanilla ASCII" or other +form. Any alternate format must include the full Project Gutenberg-tm +License as specified in paragraph 1.E.1. + +1.E.7. Do not charge a fee for access to, viewing, displaying, +performing, copying or distributing any Project Gutenberg-tm works +unless you comply with paragraph 1.E.8 or 1.E.9. + +1.E.8. You may charge a reasonable fee for copies of or providing +access to or distributing Project Gutenberg-tm electronic works provided +that + +- You pay a royalty fee of 20% of the gross profits you derive from + the use of Project Gutenberg-tm works calculated using the method + you already use to calculate your applicable taxes. The fee is + owed to the owner of the Project Gutenberg-tm trademark, but he + has agreed to donate royalties under this paragraph to the + Project Gutenberg Literary Archive Foundation. Royalty payments + must be paid within 60 days following each date on which you + prepare (or are legally required to prepare) your periodic tax + returns. Royalty payments should be clearly marked as such and + sent to the Project Gutenberg Literary Archive Foundation at the + address specified in Section 4, "Information about donations to + the Project Gutenberg Literary Archive Foundation." + +- You provide a full refund of any money paid by a user who notifies + you in writing (or by e-mail) within 30 days of receipt that s/he + does not agree to the terms of the full Project Gutenberg-tm + License. You must require such a user to return or + destroy all copies of the works possessed in a physical medium + and discontinue all use of and all access to other copies of + Project Gutenberg-tm works. + +- You provide, in accordance with paragraph 1.F.3, a full refund of any + money paid for a work or a replacement copy, if a defect in the + electronic work is discovered and reported to you within 90 days + of receipt of the work. + +- You comply with all other terms of this agreement for free + distribution of Project Gutenberg-tm works. + +1.E.9. If you wish to charge a fee or distribute a Project Gutenberg-tm +electronic work or group of works on different terms than are set +forth in this agreement, you must obtain permission in writing from +both the Project Gutenberg Literary Archive Foundation and Michael +Hart, the owner of the Project Gutenberg-tm trademark. Contact the +Foundation as set forth in Section 3 below. + +1.F. + +1.F.1. Project Gutenberg volunteers and employees expend considerable +effort to identify, do copyright research on, transcribe and proofread +public domain works in creating the Project Gutenberg-tm +collection. Despite these efforts, Project Gutenberg-tm electronic +works, and the medium on which they may be stored, may contain +"Defects," such as, but not limited to, incomplete, inaccurate or +corrupt data, transcription errors, a copyright or other intellectual +property infringement, a defective or damaged disk or other medium, a +computer virus, or computer codes that damage or cannot be read by +your equipment. + +1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right +of Replacement or Refund" described in paragraph 1.F.3, the Project +Gutenberg Literary Archive Foundation, the owner of the Project +Gutenberg-tm trademark, and any other party distributing a Project +Gutenberg-tm electronic work under this agreement, disclaim all +liability to you for damages, costs and expenses, including legal +fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT +LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE +PROVIDED IN PARAGRAPH F3. YOU AGREE THAT THE FOUNDATION, THE +TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE +LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR +INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH +DAMAGE. + +1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a +defect in this electronic work within 90 days of receiving it, you can +receive a refund of the money (if any) you paid for it by sending a +written explanation to the person you received the work from. If you +received the work on a physical medium, you must return the medium with +your written explanation. The person or entity that provided you with +the defective work may elect to provide a replacement copy in lieu of a +refund. If you received the work electronically, the person or entity +providing it to you may choose to give you a second opportunity to +receive the work electronically in lieu of a refund. If the second copy +is also defective, you may demand a refund in writing without further +opportunities to fix the problem. + +1.F.4. Except for the limited right of replacement or refund set forth +in paragraph 1.F.3, this work is provided to you 'AS-IS', WITH NO OTHER +WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO +WARRANTIES OF MERCHANTIBILITY OR FITNESS FOR ANY PURPOSE. + +1.F.5. Some states do not allow disclaimers of certain implied +warranties or the exclusion or limitation of certain types of damages. +If any disclaimer or limitation set forth in this agreement violates the +law of the state applicable to this agreement, the agreement shall be +interpreted to make the maximum disclaimer or limitation permitted by +the applicable state law. The invalidity or unenforceability of any +provision of this agreement shall not void the remaining provisions. + +1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the +trademark owner, any agent or employee of the Foundation, anyone +providing copies of Project Gutenberg-tm electronic works in accordance +with this agreement, and any volunteers associated with the production, +promotion and distribution of Project Gutenberg-tm electronic works, +harmless from all liability, costs and expenses, including legal fees, +that arise directly or indirectly from any of the following which you do +or cause to occur: (a) distribution of this or any Project Gutenberg-tm +work, (b) alteration, modification, or additions or deletions to any +Project Gutenberg-tm work, and (c) any Defect you cause. + + +Section 2. Information about the Mission of Project Gutenberg-tm + +Project Gutenberg-tm is synonymous with the free distribution of +electronic works in formats readable by the widest variety of computers +including obsolete, old, middle-aged and new computers. It exists +because of the efforts of hundreds of volunteers and donations from +people in all walks of life. + +Volunteers and financial support to provide volunteers with the +assistance they need, is critical to reaching Project Gutenberg-tm's +goals and ensuring that the Project Gutenberg-tm collection will +remain freely available for generations to come. In 2001, the Project +Gutenberg Literary Archive Foundation was created to provide a secure +and permanent future for Project Gutenberg-tm and future generations. +To learn more about the Project Gutenberg Literary Archive Foundation +and how your efforts and donations can help, see Sections 3 and 4 +and the Foundation web page at https://www.pglaf.org. + + +Section 3. Information about the Project Gutenberg Literary Archive +Foundation + +The Project Gutenberg Literary Archive Foundation is a non profit +501(c)(3) educational corporation organized under the laws of the +state of Mississippi and granted tax exempt status by the Internal +Revenue Service. The Foundation's EIN or federal tax identification +number is 64-6221541. Its 501(c)(3) letter is posted at +https://pglaf.org/fundraising. Contributions to the Project Gutenberg +Literary Archive Foundation are tax deductible to the full extent +permitted by U.S. federal laws and your state's laws. + +The Foundation's principal office is located at 4557 Melan Dr. S. +Fairbanks, AK, 99712., but its volunteers and employees are scattered +throughout numerous locations. Its business office is located at +809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887, email +business@pglaf.org. Email contact links and up to date contact +information can be found at the Foundation's web site and official +page at https://pglaf.org + +For additional contact information: + Dr. Gregory B. Newby + Chief Executive and Director + gbnewby@pglaf.org + +Section 4. Information about Donations to the Project Gutenberg +Literary Archive Foundation + +Project Gutenberg-tm depends upon and cannot survive without wide +spread public support and donations to carry out its mission of +increasing the number of public domain and licensed works that can be +freely distributed in machine readable form accessible by the widest +array of equipment including outdated equipment. Many small donations +($1 to $5,000) are particularly important to maintaining tax exempt +status with the IRS. + +The Foundation is committed to complying with the laws regulating +charities and charitable donations in all 50 states of the United +States. Compliance requirements are not uniform and it takes a +considerable effort, much paperwork and many fees to meet and keep up +with these requirements. We do not solicit donations in locations +where we have not received written confirmation of compliance. To +SEND DONATIONS or determine the status of compliance for any +particular state visit https://pglaf.org + +While we cannot and do not solicit contributions from states where we +have not met the solicitation requirements, we know of no prohibition +against accepting unsolicited donations from donors in such states who +approach us with offers to donate. + +International donations are gratefully accepted, but we cannot make +any statements concerning tax treatment of donations received from +outside the United States. U.S. laws alone swamp our small staff. + +Please check the Project Gutenberg Web pages for current donation +methods and addresses. Donations are accepted in a number of other +ways including including checks, online payments and credit card +donations. To donate, please visit: https://pglaf.org/donate + + +Section 5. General Information About Project Gutenberg-tm electronic +works. + +Professor Michael S. Hart was the originator of the Project Gutenberg-tm +concept of a library of electronic works that could be freely shared +with anyone. For thirty years, he produced and distributed Project +Gutenberg-tm eBooks with only a loose network of volunteer support. + +Project Gutenberg-tm eBooks are often created from several printed +editions, all of which are confirmed as Public Domain in the U.S. +unless a copyright notice is included. Thus, we do not necessarily +keep eBooks in compliance with any particular paper edition. + +Most people start at our Web site which has the main PG search facility: + + https://www.gutenberg.org + +This Web site includes information about Project Gutenberg-tm, +including how to make donations to the Project Gutenberg Literary +Archive Foundation, how to help produce our new eBooks, and how to +subscribe to our email newsletter to hear about new eBooks. + +*** END: FULL LICENSE *** + + + +</pre> + +</body> +</html> + |